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Pinnacle Day: 6th - 7th HCF and LCM
HCF and LCM
LCM (Least common multiple) of two or
more given numbers is the least number
that is exactly divisible by each of them.
HCF (Highest common factor) of two or
more numbers is the greatest number
that divides each of them exactly. HCF is
also known as the ‘Highest common
Divisor’ (HCD) and the Greatest Common
Measure (GCM).
The concept of multiples and factors
If X , Y , and Z are three natural ?
numbers and X × Y = Z , then
(i). X and Y are called the factors of
Z.
(ii). Z is said to be divisible by X and
Y.
(iii). Z is said to be a multiple of X
and Y.
Example: The set of positive Integers
which are factors of 18 is (1, 2, 3, 6, 9,
18).
Basic Concepts of H.C.F. and LCM
Method of finding H.C.F.
To ?nd the HCF of the given numbers
1. Break the given numbers into their
prime factors.
2. The HCF will be the product of all the
prime factors common to all the
numbers.
Let us learn the process of ?nding HCF
with the help of some solved examples.
Example:-
Find the HCF of 96, 36 and 18 ?
Solution:-
96 = 2 × 3 × 2 × 2 × 2 × 2
36 = 2 × 3 × 2 × 3
18 = 2 × 3 × 3
Therefore, the HCF of 96, 36 and 18 is
the product of the highest number of
common factors in the given numbers
i.e., 2 × 3 = 6. In other words, 6 is the
largest possible integer, which can divide
96, 36 and 18 without leaving any
remainder.
Example:- Find the H.C.F. of 42 and 70 ?
Solution:- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, H.C.F. of 42 and 70 = 2 × 7
= 14.
HCF by Division Method
Example:- HCF of 24, 48, 72, and 100.
Solution:- To start the division method
select the smallest two numbers./
HCF of 24 and 48 = 24
HCF of 24, 48 and 72 = 24
HCF of 24, 48, 72, and 100 = 4.
Example:- HCF of 1785, 1995, 3381.
Solution:-
HCF of 1785 and 1995 = 105
HCF of 1785, 1995 and 3381 = 21
NOTE :-
(i). HCF of two prime numbers is
always 1.
(ii). HCF of co-prime numbers is
always 1.
Method of finding L.C.M
The Least Common Multiple of two or
more numbers is the smallest number
which is exactly divisible by all of them.
In other words, it is the product of the
highest powers of all the prime factors of
the given numbers.
To ?nd the LCM of given numbers:
1. Break the given numbers into
their prime factors.
2. The LCM will be the product of
the highest power of all the
factors that occur in the given
numbers.
Let us take some solved examples.
Example:- Find the LCM of 96, 36 and 18.
Solution : 96 = 2 × 2 × 2 × 2 × 2 × 3
= × ; 36 = 2 × 2 × 3 × 3 = × 2
5
3
1
2
2
3
2
18 = 2 × 3 × 3 = × 2
1
3
2
Therefore, LCM of 96, 36 and 18 is the
product of the highest powers of all the
prime factors, i.e. 2
5
× 3
2
= 32 × 9 = 288
That is, 288 is the smallest integer which
is divisible by 96, 36 and 18 without
leaving any remainder.
Example:- Find the LCM of 42 and 70
Solution :- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, LCM is 2 × 3 × 5 × 7 = 210.
Example:- LCM of 6, 12, 8 ?
Solution :-
LCM = 2 × 2 × 3 × 2 = 24
HCF of 6, 12, 18
Firstly ?nd out the factors of 6, 12, 18
and then multiply the common factors.
6 = , 12 = , 18 = 2 × 3 2 × 2 × 3 2 × 3 × 3
HCF = 2 × 3 = 6
Try ?nding HCF and LCM of 3, 6, 9, 12
yourself. HCF of 3, 6, 9, 12 can also be
found by Division method. It is useful
when the numbers are bigger.
NOTE :-
(i). HCF of A, B and C is the highest
divisor which can exactly divide
A, B, and C.
(ii). LCM of A, B and C is the lowest
dividend which is exactly
divisible by A, B, and C.
There is one very important relationship,
given below, between two numbers and
their HCF and LCM. Many problems have
appeared in various competitive exams
based on this relationship.
Important Concepts:-
(1).
LCM × HCF = 1st number × 2nd number
Example:- For numbers 8 and 12,
LCM = 24 and HCF = 4
Now, LCM HCF = 24 4 = 96 × ×
also, 8 12 = 96 ×
(2). HCF of some numbers is always a
factor of LCM of the numbers.
(3).
LCM of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
(4).
HCF of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
Example: LCM and HCF of , and
1
2
2
3
3
4
Solution:- LCM =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
6
1
HCF =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
1
12
(5) Co- Prime numbers :
If the HCF of two numbers is 1 then they
are called co-prime numbers.
(6) = ;
??????
??????
??????????????
where LCM and HCF are of two numbers
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Page 2
Pinnacle Day: 6th - 7th HCF and LCM
HCF and LCM
LCM (Least common multiple) of two or
more given numbers is the least number
that is exactly divisible by each of them.
HCF (Highest common factor) of two or
more numbers is the greatest number
that divides each of them exactly. HCF is
also known as the ‘Highest common
Divisor’ (HCD) and the Greatest Common
Measure (GCM).
The concept of multiples and factors
If X , Y , and Z are three natural ?
numbers and X × Y = Z , then
(i). X and Y are called the factors of
Z.
(ii). Z is said to be divisible by X and
Y.
(iii). Z is said to be a multiple of X
and Y.
Example: The set of positive Integers
which are factors of 18 is (1, 2, 3, 6, 9,
18).
Basic Concepts of H.C.F. and LCM
Method of finding H.C.F.
To ?nd the HCF of the given numbers
1. Break the given numbers into their
prime factors.
2. The HCF will be the product of all the
prime factors common to all the
numbers.
Let us learn the process of ?nding HCF
with the help of some solved examples.
Example:-
Find the HCF of 96, 36 and 18 ?
Solution:-
96 = 2 × 3 × 2 × 2 × 2 × 2
36 = 2 × 3 × 2 × 3
18 = 2 × 3 × 3
Therefore, the HCF of 96, 36 and 18 is
the product of the highest number of
common factors in the given numbers
i.e., 2 × 3 = 6. In other words, 6 is the
largest possible integer, which can divide
96, 36 and 18 without leaving any
remainder.
Example:- Find the H.C.F. of 42 and 70 ?
Solution:- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, H.C.F. of 42 and 70 = 2 × 7
= 14.
HCF by Division Method
Example:- HCF of 24, 48, 72, and 100.
Solution:- To start the division method
select the smallest two numbers./
HCF of 24 and 48 = 24
HCF of 24, 48 and 72 = 24
HCF of 24, 48, 72, and 100 = 4.
Example:- HCF of 1785, 1995, 3381.
Solution:-
HCF of 1785 and 1995 = 105
HCF of 1785, 1995 and 3381 = 21
NOTE :-
(i). HCF of two prime numbers is
always 1.
(ii). HCF of co-prime numbers is
always 1.
Method of finding L.C.M
The Least Common Multiple of two or
more numbers is the smallest number
which is exactly divisible by all of them.
In other words, it is the product of the
highest powers of all the prime factors of
the given numbers.
To ?nd the LCM of given numbers:
1. Break the given numbers into
their prime factors.
2. The LCM will be the product of
the highest power of all the
factors that occur in the given
numbers.
Let us take some solved examples.
Example:- Find the LCM of 96, 36 and 18.
Solution : 96 = 2 × 2 × 2 × 2 × 2 × 3
= × ; 36 = 2 × 2 × 3 × 3 = × 2
5
3
1
2
2
3
2
18 = 2 × 3 × 3 = × 2
1
3
2
Therefore, LCM of 96, 36 and 18 is the
product of the highest powers of all the
prime factors, i.e. 2
5
× 3
2
= 32 × 9 = 288
That is, 288 is the smallest integer which
is divisible by 96, 36 and 18 without
leaving any remainder.
Example:- Find the LCM of 42 and 70
Solution :- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, LCM is 2 × 3 × 5 × 7 = 210.
Example:- LCM of 6, 12, 8 ?
Solution :-
LCM = 2 × 2 × 3 × 2 = 24
HCF of 6, 12, 18
Firstly ?nd out the factors of 6, 12, 18
and then multiply the common factors.
6 = , 12 = , 18 = 2 × 3 2 × 2 × 3 2 × 3 × 3
HCF = 2 × 3 = 6
Try ?nding HCF and LCM of 3, 6, 9, 12
yourself. HCF of 3, 6, 9, 12 can also be
found by Division method. It is useful
when the numbers are bigger.
NOTE :-
(i). HCF of A, B and C is the highest
divisor which can exactly divide
A, B, and C.
(ii). LCM of A, B and C is the lowest
dividend which is exactly
divisible by A, B, and C.
There is one very important relationship,
given below, between two numbers and
their HCF and LCM. Many problems have
appeared in various competitive exams
based on this relationship.
Important Concepts:-
(1).
LCM × HCF = 1st number × 2nd number
Example:- For numbers 8 and 12,
LCM = 24 and HCF = 4
Now, LCM HCF = 24 4 = 96 × ×
also, 8 12 = 96 ×
(2). HCF of some numbers is always a
factor of LCM of the numbers.
(3).
LCM of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
(4).
HCF of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
Example: LCM and HCF of , and
1
2
2
3
3
4
Solution:- LCM =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
6
1
HCF =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
1
12
(5) Co- Prime numbers :
If the HCF of two numbers is 1 then they
are called co-prime numbers.
(6) = ;
??????
??????
??????????????
where LCM and HCF are of two numbers
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Pinnacle Day: 6th - 7th HCF and LCM
. If we ?nd two co-prime ??
1
?????? ??
2
factors, , of the Product ??
1
?????? ??
2
as obtained above.
(7)
Example:-
(8).
If 1st number = and 2nd number = ??
1
??
2
HCF of = Hx ??
1
HCF of = Hy ??
2
And HCF of and = H ??
1
??
2
So ,
Difference between and = Hx - Hy ??
1
??
2
= H(x - y)
Example :-
= 24 and = 36 ??
1
??
2
Note :- HCF is always either the
difference of two numbers or factors of
difference of two numbers.
(9). When the second divisor is a factor
of the ?rst divisor, then the second
remainder is obtained by dividing the ?rst
remainder by the second divisor.
Example :- When 29 is divided by 8, the
remainder obtained is 5, then what will
be the remainder when the same number
is divided by 4 ?
Solution:- Here, the second divisor that is
4 is a factor of the ?rst divisor that is 8.
So, on dividing the ?rst remainder that is
5 by the second divisor that is 4, we get
our second remainder which is 1. So, the
required answer is 1.
Variety Questions
Q.1. The LCM of x² - 8x + 15 and x² - 5x + 6
is:
SSC CPO 05/10/2023 (2nd Shift)
(a) (x + 5)(x + 2)(x + 3)
(b) (x - 5)(x - 2)(x - 3)
(c) (x + 5)(x - 2)(x - 3)
(d) (x - 2)(x - 3)
2
(x - 5)
Q.2. The greatest possible length that
can be used to measure exactly the
lengths of 3 m 15 cm, 5 m, and 6 m 85
cm is:
SSC CPO 05/10/2023 (2nd Shift)
(a) 11 cm (b) 7 cm (c) 9 cm (d) 5 cm
Q.3. What is the HCF of ( + 1) and ( - 1) ??
6
??
4
SSC CPO 05/10/2023 (1st Shift)
(a) (1 + ) (b) (1 + x) (c) 1 (d) (1 ) ??
2
- ??
2
Q.4. The LCM of two prime numbers x
and y (x > y) is 533. The value of 4y - x is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 11 (b) 21 (c) 18 (d) 23
Q.5. What is the HCF of the polynomials
( ³ - 8), ( ³ - 6 ² + 12 - 8) and ( ³ - 4 ² + 4 ?? ?? ?? ?? ?? ??
) ? ??
SSC MTS 12/09/2023 (3rd Shift)
(a) ( - 1) (b) ( - 2) (c) ( - 8) (d) ( - 4) ?? ?? ?? ??
Q.6. Find the greatest two digit number
which on dividing 219, 365, 511 leaves
the remainders 3, 5, 7, respectively.
SSC MTS 01/09/2023 (3rd Shift)
(a) 63 (b) 82 (c) 53 (d) 72
Q.7. There is a circular path around a
sports ?eld. Rahul takes 15 minutes to
drive one round of the ?eld, while Anil
takes 18 minutes for the same. Suppose
they both start from the same point and
at the same time, and go in the same
direction. After how many minutes will
they meet again at the starting point?
SSC MTS 11/05/2023 (Afternoon)
(a) 120 (b) 100 (c) 80 (d) 90
Q.8. Let x = 224 and, y = 322. If the
highest common factor of 23x and a × y
is divisible by x and y, then what can be
the possible value of a?
SSC CPO 11/11/2022 (Evening)
(a) 16 (b) 8 (c) 12 (d) 4
Q.9. The highest common factor of 108,
72 and 5a is a. What can be the least
common multiple of 108, 72 and a ?
SSC CPO 10/11/2022 (Afternoon)
(a) 432 (b) 324 (c) 108 (d) 216
Q.10. What is the greatest positive
integer that divides 554, 714 and 213
leaving the remainder 43, 57 and 67,
respectively?
SSC CPO 10/11/2022 (Afternoon)
(a) 95 (b) 71 (c) 83 (d) 73
Q.11. Find the HCF of ( – 1) and 4
315
( – 1). 4
25
SSC MTS 26/07/2022 (Morning)
(a) 1 (b) ( – 1) (c) 1024 (d) 1023 4
25
Q.12. 120 apples, 240 oranges and 150
pears are packed in cartons in such a
way that each carton has the same
number of fruits, each carton contains
only one type of fruit and fruit is left
unpacked. What is the smallest possible
number of cartons needed for the
purpose ?
SSC MTS 08/07/2022 (Evening)
(a) 50 (b) 40 (c) 17 (d) 30
Q.13. 13, a, b, c are four distinct numbers
and the HCF of each pair of numbers (13,
a) : (13, b) : (13, c) is 13, where a, b, c are
each less than 60 and a < b < c. What is
the value of ?
?? + ??
??
SSC CGL 13/04/2021 (Morning)
(a) 3.5 (b) 2 (c) 5 (d) 4.5
Q.14. The sum of two numbers is 1215
and their HCF is 81. If the numbers lie
between 500 and 700, then the sum of
reciprocals of the numbers is
SSC CPO 13/12/2019 (Evening)
(a) (b) (c) (d)
5
1512
5
378
5
702
5
1188
Q.15. If r is the remainder when each of
6454, 7306 and 8797 is divided by the
greatest number d (d > 1), then (d - r) is:
SSC CPO 13/12/2019 (Morning)
(a) 126 (b) 64 (c) 137 (d) 149
Q.16. In ?nding the HCF of two numbers
by division method, the quotients are 1, 8
and 2 respectively, and the last divisor is
105, what is the sum of the numbers?
SSC CPO 11/12/2019 (Evening)
(a) 3570 (b) 3885 (c) 3780 (d) 3675
Q.17. When the smallest number x is
divided by 5, 6, 8, 9 and 12, it gives
remainder 1 in each case. But x is
divisible by 13. What will be the
remainder when x will be divided by 31
SSC MTS 20/08/2019 (Afternoon)
(a) 1 (b) 5 (c) 3 (d) 0
Q.18. The Highest Common Factor and
Lowest Common Multiple of two
numbers p and q are A and B
respectively. IF A + B = p + q, then the
value of is : ??
3
+ ??
3
SSC MTS 09/08/2019 (Evening)
(a) p
3
(b) q
3
(c) p
3
+ q
3
(d) p
3
- q
3
Q.19. What is the largest number that
divides 460, 491 and 553 and leaves the
remainder 26 in each case ?
SSC MTS 06/08/2019 (Afternoon)
(a) 27 (b)35 (c) 33 (d) 31
Q.20. A is the smallest three-digit
number which when divided by 3, 4 and 5
gives remainder 1, 2 and 3 respectively.
What is the sum of the digits of A ?
SSC MTS 05/08/2019 (Afternoon)
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Page 3
Pinnacle Day: 6th - 7th HCF and LCM
HCF and LCM
LCM (Least common multiple) of two or
more given numbers is the least number
that is exactly divisible by each of them.
HCF (Highest common factor) of two or
more numbers is the greatest number
that divides each of them exactly. HCF is
also known as the ‘Highest common
Divisor’ (HCD) and the Greatest Common
Measure (GCM).
The concept of multiples and factors
If X , Y , and Z are three natural ?
numbers and X × Y = Z , then
(i). X and Y are called the factors of
Z.
(ii). Z is said to be divisible by X and
Y.
(iii). Z is said to be a multiple of X
and Y.
Example: The set of positive Integers
which are factors of 18 is (1, 2, 3, 6, 9,
18).
Basic Concepts of H.C.F. and LCM
Method of finding H.C.F.
To ?nd the HCF of the given numbers
1. Break the given numbers into their
prime factors.
2. The HCF will be the product of all the
prime factors common to all the
numbers.
Let us learn the process of ?nding HCF
with the help of some solved examples.
Example:-
Find the HCF of 96, 36 and 18 ?
Solution:-
96 = 2 × 3 × 2 × 2 × 2 × 2
36 = 2 × 3 × 2 × 3
18 = 2 × 3 × 3
Therefore, the HCF of 96, 36 and 18 is
the product of the highest number of
common factors in the given numbers
i.e., 2 × 3 = 6. In other words, 6 is the
largest possible integer, which can divide
96, 36 and 18 without leaving any
remainder.
Example:- Find the H.C.F. of 42 and 70 ?
Solution:- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, H.C.F. of 42 and 70 = 2 × 7
= 14.
HCF by Division Method
Example:- HCF of 24, 48, 72, and 100.
Solution:- To start the division method
select the smallest two numbers./
HCF of 24 and 48 = 24
HCF of 24, 48 and 72 = 24
HCF of 24, 48, 72, and 100 = 4.
Example:- HCF of 1785, 1995, 3381.
Solution:-
HCF of 1785 and 1995 = 105
HCF of 1785, 1995 and 3381 = 21
NOTE :-
(i). HCF of two prime numbers is
always 1.
(ii). HCF of co-prime numbers is
always 1.
Method of finding L.C.M
The Least Common Multiple of two or
more numbers is the smallest number
which is exactly divisible by all of them.
In other words, it is the product of the
highest powers of all the prime factors of
the given numbers.
To ?nd the LCM of given numbers:
1. Break the given numbers into
their prime factors.
2. The LCM will be the product of
the highest power of all the
factors that occur in the given
numbers.
Let us take some solved examples.
Example:- Find the LCM of 96, 36 and 18.
Solution : 96 = 2 × 2 × 2 × 2 × 2 × 3
= × ; 36 = 2 × 2 × 3 × 3 = × 2
5
3
1
2
2
3
2
18 = 2 × 3 × 3 = × 2
1
3
2
Therefore, LCM of 96, 36 and 18 is the
product of the highest powers of all the
prime factors, i.e. 2
5
× 3
2
= 32 × 9 = 288
That is, 288 is the smallest integer which
is divisible by 96, 36 and 18 without
leaving any remainder.
Example:- Find the LCM of 42 and 70
Solution :- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, LCM is 2 × 3 × 5 × 7 = 210.
Example:- LCM of 6, 12, 8 ?
Solution :-
LCM = 2 × 2 × 3 × 2 = 24
HCF of 6, 12, 18
Firstly ?nd out the factors of 6, 12, 18
and then multiply the common factors.
6 = , 12 = , 18 = 2 × 3 2 × 2 × 3 2 × 3 × 3
HCF = 2 × 3 = 6
Try ?nding HCF and LCM of 3, 6, 9, 12
yourself. HCF of 3, 6, 9, 12 can also be
found by Division method. It is useful
when the numbers are bigger.
NOTE :-
(i). HCF of A, B and C is the highest
divisor which can exactly divide
A, B, and C.
(ii). LCM of A, B and C is the lowest
dividend which is exactly
divisible by A, B, and C.
There is one very important relationship,
given below, between two numbers and
their HCF and LCM. Many problems have
appeared in various competitive exams
based on this relationship.
Important Concepts:-
(1).
LCM × HCF = 1st number × 2nd number
Example:- For numbers 8 and 12,
LCM = 24 and HCF = 4
Now, LCM HCF = 24 4 = 96 × ×
also, 8 12 = 96 ×
(2). HCF of some numbers is always a
factor of LCM of the numbers.
(3).
LCM of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
(4).
HCF of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
Example: LCM and HCF of , and
1
2
2
3
3
4
Solution:- LCM =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
6
1
HCF =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
1
12
(5) Co- Prime numbers :
If the HCF of two numbers is 1 then they
are called co-prime numbers.
(6) = ;
??????
??????
??????????????
where LCM and HCF are of two numbers
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Pinnacle Day: 6th - 7th HCF and LCM
. If we ?nd two co-prime ??
1
?????? ??
2
factors, , of the Product ??
1
?????? ??
2
as obtained above.
(7)
Example:-
(8).
If 1st number = and 2nd number = ??
1
??
2
HCF of = Hx ??
1
HCF of = Hy ??
2
And HCF of and = H ??
1
??
2
So ,
Difference between and = Hx - Hy ??
1
??
2
= H(x - y)
Example :-
= 24 and = 36 ??
1
??
2
Note :- HCF is always either the
difference of two numbers or factors of
difference of two numbers.
(9). When the second divisor is a factor
of the ?rst divisor, then the second
remainder is obtained by dividing the ?rst
remainder by the second divisor.
Example :- When 29 is divided by 8, the
remainder obtained is 5, then what will
be the remainder when the same number
is divided by 4 ?
Solution:- Here, the second divisor that is
4 is a factor of the ?rst divisor that is 8.
So, on dividing the ?rst remainder that is
5 by the second divisor that is 4, we get
our second remainder which is 1. So, the
required answer is 1.
Variety Questions
Q.1. The LCM of x² - 8x + 15 and x² - 5x + 6
is:
SSC CPO 05/10/2023 (2nd Shift)
(a) (x + 5)(x + 2)(x + 3)
(b) (x - 5)(x - 2)(x - 3)
(c) (x + 5)(x - 2)(x - 3)
(d) (x - 2)(x - 3)
2
(x - 5)
Q.2. The greatest possible length that
can be used to measure exactly the
lengths of 3 m 15 cm, 5 m, and 6 m 85
cm is:
SSC CPO 05/10/2023 (2nd Shift)
(a) 11 cm (b) 7 cm (c) 9 cm (d) 5 cm
Q.3. What is the HCF of ( + 1) and ( - 1) ??
6
??
4
SSC CPO 05/10/2023 (1st Shift)
(a) (1 + ) (b) (1 + x) (c) 1 (d) (1 ) ??
2
- ??
2
Q.4. The LCM of two prime numbers x
and y (x > y) is 533. The value of 4y - x is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 11 (b) 21 (c) 18 (d) 23
Q.5. What is the HCF of the polynomials
( ³ - 8), ( ³ - 6 ² + 12 - 8) and ( ³ - 4 ² + 4 ?? ?? ?? ?? ?? ??
) ? ??
SSC MTS 12/09/2023 (3rd Shift)
(a) ( - 1) (b) ( - 2) (c) ( - 8) (d) ( - 4) ?? ?? ?? ??
Q.6. Find the greatest two digit number
which on dividing 219, 365, 511 leaves
the remainders 3, 5, 7, respectively.
SSC MTS 01/09/2023 (3rd Shift)
(a) 63 (b) 82 (c) 53 (d) 72
Q.7. There is a circular path around a
sports ?eld. Rahul takes 15 minutes to
drive one round of the ?eld, while Anil
takes 18 minutes for the same. Suppose
they both start from the same point and
at the same time, and go in the same
direction. After how many minutes will
they meet again at the starting point?
SSC MTS 11/05/2023 (Afternoon)
(a) 120 (b) 100 (c) 80 (d) 90
Q.8. Let x = 224 and, y = 322. If the
highest common factor of 23x and a × y
is divisible by x and y, then what can be
the possible value of a?
SSC CPO 11/11/2022 (Evening)
(a) 16 (b) 8 (c) 12 (d) 4
Q.9. The highest common factor of 108,
72 and 5a is a. What can be the least
common multiple of 108, 72 and a ?
SSC CPO 10/11/2022 (Afternoon)
(a) 432 (b) 324 (c) 108 (d) 216
Q.10. What is the greatest positive
integer that divides 554, 714 and 213
leaving the remainder 43, 57 and 67,
respectively?
SSC CPO 10/11/2022 (Afternoon)
(a) 95 (b) 71 (c) 83 (d) 73
Q.11. Find the HCF of ( – 1) and 4
315
( – 1). 4
25
SSC MTS 26/07/2022 (Morning)
(a) 1 (b) ( – 1) (c) 1024 (d) 1023 4
25
Q.12. 120 apples, 240 oranges and 150
pears are packed in cartons in such a
way that each carton has the same
number of fruits, each carton contains
only one type of fruit and fruit is left
unpacked. What is the smallest possible
number of cartons needed for the
purpose ?
SSC MTS 08/07/2022 (Evening)
(a) 50 (b) 40 (c) 17 (d) 30
Q.13. 13, a, b, c are four distinct numbers
and the HCF of each pair of numbers (13,
a) : (13, b) : (13, c) is 13, where a, b, c are
each less than 60 and a < b < c. What is
the value of ?
?? + ??
??
SSC CGL 13/04/2021 (Morning)
(a) 3.5 (b) 2 (c) 5 (d) 4.5
Q.14. The sum of two numbers is 1215
and their HCF is 81. If the numbers lie
between 500 and 700, then the sum of
reciprocals of the numbers is
SSC CPO 13/12/2019 (Evening)
(a) (b) (c) (d)
5
1512
5
378
5
702
5
1188
Q.15. If r is the remainder when each of
6454, 7306 and 8797 is divided by the
greatest number d (d > 1), then (d - r) is:
SSC CPO 13/12/2019 (Morning)
(a) 126 (b) 64 (c) 137 (d) 149
Q.16. In ?nding the HCF of two numbers
by division method, the quotients are 1, 8
and 2 respectively, and the last divisor is
105, what is the sum of the numbers?
SSC CPO 11/12/2019 (Evening)
(a) 3570 (b) 3885 (c) 3780 (d) 3675
Q.17. When the smallest number x is
divided by 5, 6, 8, 9 and 12, it gives
remainder 1 in each case. But x is
divisible by 13. What will be the
remainder when x will be divided by 31
SSC MTS 20/08/2019 (Afternoon)
(a) 1 (b) 5 (c) 3 (d) 0
Q.18. The Highest Common Factor and
Lowest Common Multiple of two
numbers p and q are A and B
respectively. IF A + B = p + q, then the
value of is : ??
3
+ ??
3
SSC MTS 09/08/2019 (Evening)
(a) p
3
(b) q
3
(c) p
3
+ q
3
(d) p
3
- q
3
Q.19. What is the largest number that
divides 460, 491 and 553 and leaves the
remainder 26 in each case ?
SSC MTS 06/08/2019 (Afternoon)
(a) 27 (b)35 (c) 33 (d) 31
Q.20. A is the smallest three-digit
number which when divided by 3, 4 and 5
gives remainder 1, 2 and 3 respectively.
What is the sum of the digits of A ?
SSC MTS 05/08/2019 (Afternoon)
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 28
Pinnacle Day: 6th - 7th HCF and LCM
(a) 11 (b) 10 (c) 6 (d) 8
Q.21. The product of two numbers is
6760 and their HCF is 13. How many
such pairs of numbers can be formed?
SSC CPO 16/03/2019 (Evening)
(a) 2 (b) 3 (c) 1 (d) 4
Q.22. An oil merchant has 3 varieties of
oil of volumes 432, 594 and 702
respectively. The number of cans of
equal size that would be required to ?ll
the oil separately is:
SSC CPO 16/03/2019 (Afternoon)
(a) 13, 15, 17 (b) 8, 11, 13
(c) 8, 13, 15 (d) 6, 9, 11
Q.23. Which of the following statements
is true ?
SSC CPO 14/03/2019 (Morning)
(a) LCM of two natural numbers is
divisible by their HCF.
(b) HCF + LCM of two numbers =
Product of the two numbers.
(c) Two prime numbers are co-prime
numbers if their LCM is 1.
(d) HCF of two numbers is the smallest
common divisor of both numbers.
Practice Questions
SSC CPO 2023 Tier - 1
Q.24. The largest number of four digits
that is exactly divisible by 17 and 36 is:
SSC CPO 03/10/2023 (1st Shift)
(a) 8568 (b) 9180 (c) 9792 (d) 7956
Q.25. Two numbers are in the ratio 3 : 4.
The product of their HCF and LCM is
2700. The sum of the numbers is:
SSC CPO 03/10/2023 (1st Shift)
(a) 45 (b) 105 (c) 60 (d) 15
Q.26. Find the greatest number which
when divides 261, 853 and 1221, leaves a
remainder of 5 in each case.
SSC CPO 03/10/2023 (2nd Shift)
(a) 18 (b) 17 (c) 16 (d) 19
Q.27. Find the LCM of , and .
3
2
81
16
9
8
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
111
2
91
2
81
2
101
2
Q.28. Find the greatest possible length
(in metres) that can be used to exactly
measure the lengths 6 m, 5 m 25 cm and
12 m 50 cm.
SSC CPO 03/10/2023 (2nd Shift)
(a) 0.35 m (b) 0.90 m (c) 0.75 m (d) 0.25 m
Q.29. The least number which should be
added to 1351 so that the sum is exactly
divisible by 2, 4, 6 and 8 is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 13 (b) 11 (c) 15 (d) 17
Q.30. Two numbers are in the ratio of 4:
3. The product of their HCF and LCM is
2700. The difference between the
numbers is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 25 (b) 30 (c) 15 (d) 105
Q.31. Let the HCF of m and n be 'a' and
let n = ab, LCM of m and n is given by:
SSC CPO 04/10/2023 (1st Shift)
(a) ab (b) am (c) bm (d) mn
Q.32. The LCM and HCF of two numbers
are 1105 and 5. If the LCM is 17 times
the ?rst number, then ?nd the two
numbers.
SSC CPO 04/10/2023 (2nd Shift)
(a) 55 and 85 (b) 65 and 75
(c) 60 and 80 (d) 65 and 85
Q.33. Which is the smallest natural
number that is exactly divisible by each
of 96, 108 and 144 ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 1728 (b) 864 (c) 1296 (d) 2592
Q.34. The HCF of 3888 and 3969 is:
SSC CPO 04/10/2023 (2nd Shift)
(a) 81 (b) 73 (c) 83 (d) 71
Q.35. A person has three iron bars whose
lengths are 20, 30 and 40 metres,
respectively. He wants to cut pieces of
the same length from each of the three
bars. What is the least number of total
pieces if he cuts without any wastage ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 9 (b) 10 (c) 8 (d) 11
Q.36. If the HCF of 45 and 55 is
expressible in the form of 55 × 5 + 45m,
then what is the value of m ?
SSC CPO 04/10/2023 (3rd Shift)
(a) 5 (b) –6 (c) –5 (d) 6
Q.37. Find the least number which when
divided by 4, 9, 12 and 15, leaves the
remainder 3 in each case.
SSC CPO 04/10/2023 (3rd Shift)
(a) 193 (b) 183 (c) 360 (d) 180
Q.38. Determine the LCM of two
numbers if their HCF is 12 and their ratio
is 13 : 15.
SSC CPO 04/10/2023 (3rd Shift)
(a) 2450 (b) 1780 (c) 1890 (d) 2340
Q.39. What is the smallest perfect
square number which is completely
divisible by 4, 6, 9, 12 and 15?
SSC CPO 04/10/2023 (3rd Shift)
(a) 900 (b) 961 (c) 784 (d) 841
Q.40. Which of the following numbers
leaves the remainder equal to the highest
common factor of 6, 8 and 9, when
divided by 6, 8 and 9?
SSC CPO 04/10/2023 (3rd Shift)
(a) 506 (b) 575 (c) 291 (d) 433
Q.41. Find the least number which is
exactly divisible by 20, 28, 34, 60 and 75.
SSC CPO 05/10/2023 (1st Shift)
(a) 34500 (b) 35900 (c) 35700 (d) 36220
Q.42. Three numbers are in the ratio of 5
: 7 : 9 and their LCM is 34,650. Their HCF
is:
SSC CPO 05/10/2023 (1st Shift)
(a) 110 (b) 315 (c) 99 (d) 55
Q.43. Which is the largest number that
divides each of 1036, 1813 and 3885
without leaving any remainder ?
SSC CPO 05/10/2023 (1st Shift)
(a) 259 (b) 111 (c) 333 (d) 37
Q.44. What is the LCM of 0.15, 0.18 and
0.45 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 0.6 (b) 0.9 (c) 0.81 (d) 0.09
Q.45. What will be the least number
which when doubled will be exactly
divisible by 12, 14, 16 and 18 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 636 (b) 226 (c) 428 (d) 504
Q.46. The LCM of two numbers is ?ve
times their HCF. If the product of the two
numbers is 20480, then ?nd their HCF
and LCM, respectively.
SSC CPO 05/10/2023 (3rd Shift)
(a) 64 and 320 (b) 56 and 280
(c) 48 and 240 (d) 46 and 230
Q.47. If the sum of two numbers is 60
and their HCF and LCM are 5 and 60,
respectively, then the sum of the
reciprocals of the numbers will be:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) (c) (d)
1
4
1
5
1
11
1
6
Q.48. Which of the following is the
greatest four-digit number that is
divisible by 15, 25, 40, and 75?
SSC CPO 05/10/2023 (3rd Shift)
(a) 9000 (b) 9600 (c) 9500 (d) 9200
Q.49. The least number of ?ve digits
which is exactly divisible by 9, 12, 15, 25
and 27 is:
SSC CPO 05/10/2023 (3rd Shift)
(a) 10250 (b) 10800 (c) 10600 (d) 10700
SSC MTS 2023 Tier - 1
Q.50. Three measuring tapes are 64 cm
72 cm and 96 cm, respectively. What is
the least length that can be measured by
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Page 4
Pinnacle Day: 6th - 7th HCF and LCM
HCF and LCM
LCM (Least common multiple) of two or
more given numbers is the least number
that is exactly divisible by each of them.
HCF (Highest common factor) of two or
more numbers is the greatest number
that divides each of them exactly. HCF is
also known as the ‘Highest common
Divisor’ (HCD) and the Greatest Common
Measure (GCM).
The concept of multiples and factors
If X , Y , and Z are three natural ?
numbers and X × Y = Z , then
(i). X and Y are called the factors of
Z.
(ii). Z is said to be divisible by X and
Y.
(iii). Z is said to be a multiple of X
and Y.
Example: The set of positive Integers
which are factors of 18 is (1, 2, 3, 6, 9,
18).
Basic Concepts of H.C.F. and LCM
Method of finding H.C.F.
To ?nd the HCF of the given numbers
1. Break the given numbers into their
prime factors.
2. The HCF will be the product of all the
prime factors common to all the
numbers.
Let us learn the process of ?nding HCF
with the help of some solved examples.
Example:-
Find the HCF of 96, 36 and 18 ?
Solution:-
96 = 2 × 3 × 2 × 2 × 2 × 2
36 = 2 × 3 × 2 × 3
18 = 2 × 3 × 3
Therefore, the HCF of 96, 36 and 18 is
the product of the highest number of
common factors in the given numbers
i.e., 2 × 3 = 6. In other words, 6 is the
largest possible integer, which can divide
96, 36 and 18 without leaving any
remainder.
Example:- Find the H.C.F. of 42 and 70 ?
Solution:- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, H.C.F. of 42 and 70 = 2 × 7
= 14.
HCF by Division Method
Example:- HCF of 24, 48, 72, and 100.
Solution:- To start the division method
select the smallest two numbers./
HCF of 24 and 48 = 24
HCF of 24, 48 and 72 = 24
HCF of 24, 48, 72, and 100 = 4.
Example:- HCF of 1785, 1995, 3381.
Solution:-
HCF of 1785 and 1995 = 105
HCF of 1785, 1995 and 3381 = 21
NOTE :-
(i). HCF of two prime numbers is
always 1.
(ii). HCF of co-prime numbers is
always 1.
Method of finding L.C.M
The Least Common Multiple of two or
more numbers is the smallest number
which is exactly divisible by all of them.
In other words, it is the product of the
highest powers of all the prime factors of
the given numbers.
To ?nd the LCM of given numbers:
1. Break the given numbers into
their prime factors.
2. The LCM will be the product of
the highest power of all the
factors that occur in the given
numbers.
Let us take some solved examples.
Example:- Find the LCM of 96, 36 and 18.
Solution : 96 = 2 × 2 × 2 × 2 × 2 × 3
= × ; 36 = 2 × 2 × 3 × 3 = × 2
5
3
1
2
2
3
2
18 = 2 × 3 × 3 = × 2
1
3
2
Therefore, LCM of 96, 36 and 18 is the
product of the highest powers of all the
prime factors, i.e. 2
5
× 3
2
= 32 × 9 = 288
That is, 288 is the smallest integer which
is divisible by 96, 36 and 18 without
leaving any remainder.
Example:- Find the LCM of 42 and 70
Solution :- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, LCM is 2 × 3 × 5 × 7 = 210.
Example:- LCM of 6, 12, 8 ?
Solution :-
LCM = 2 × 2 × 3 × 2 = 24
HCF of 6, 12, 18
Firstly ?nd out the factors of 6, 12, 18
and then multiply the common factors.
6 = , 12 = , 18 = 2 × 3 2 × 2 × 3 2 × 3 × 3
HCF = 2 × 3 = 6
Try ?nding HCF and LCM of 3, 6, 9, 12
yourself. HCF of 3, 6, 9, 12 can also be
found by Division method. It is useful
when the numbers are bigger.
NOTE :-
(i). HCF of A, B and C is the highest
divisor which can exactly divide
A, B, and C.
(ii). LCM of A, B and C is the lowest
dividend which is exactly
divisible by A, B, and C.
There is one very important relationship,
given below, between two numbers and
their HCF and LCM. Many problems have
appeared in various competitive exams
based on this relationship.
Important Concepts:-
(1).
LCM × HCF = 1st number × 2nd number
Example:- For numbers 8 and 12,
LCM = 24 and HCF = 4
Now, LCM HCF = 24 4 = 96 × ×
also, 8 12 = 96 ×
(2). HCF of some numbers is always a
factor of LCM of the numbers.
(3).
LCM of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
(4).
HCF of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
Example: LCM and HCF of , and
1
2
2
3
3
4
Solution:- LCM =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
6
1
HCF =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
1
12
(5) Co- Prime numbers :
If the HCF of two numbers is 1 then they
are called co-prime numbers.
(6) = ;
??????
??????
??????????????
where LCM and HCF are of two numbers
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Pinnacle Day: 6th - 7th HCF and LCM
. If we ?nd two co-prime ??
1
?????? ??
2
factors, , of the Product ??
1
?????? ??
2
as obtained above.
(7)
Example:-
(8).
If 1st number = and 2nd number = ??
1
??
2
HCF of = Hx ??
1
HCF of = Hy ??
2
And HCF of and = H ??
1
??
2
So ,
Difference between and = Hx - Hy ??
1
??
2
= H(x - y)
Example :-
= 24 and = 36 ??
1
??
2
Note :- HCF is always either the
difference of two numbers or factors of
difference of two numbers.
(9). When the second divisor is a factor
of the ?rst divisor, then the second
remainder is obtained by dividing the ?rst
remainder by the second divisor.
Example :- When 29 is divided by 8, the
remainder obtained is 5, then what will
be the remainder when the same number
is divided by 4 ?
Solution:- Here, the second divisor that is
4 is a factor of the ?rst divisor that is 8.
So, on dividing the ?rst remainder that is
5 by the second divisor that is 4, we get
our second remainder which is 1. So, the
required answer is 1.
Variety Questions
Q.1. The LCM of x² - 8x + 15 and x² - 5x + 6
is:
SSC CPO 05/10/2023 (2nd Shift)
(a) (x + 5)(x + 2)(x + 3)
(b) (x - 5)(x - 2)(x - 3)
(c) (x + 5)(x - 2)(x - 3)
(d) (x - 2)(x - 3)
2
(x - 5)
Q.2. The greatest possible length that
can be used to measure exactly the
lengths of 3 m 15 cm, 5 m, and 6 m 85
cm is:
SSC CPO 05/10/2023 (2nd Shift)
(a) 11 cm (b) 7 cm (c) 9 cm (d) 5 cm
Q.3. What is the HCF of ( + 1) and ( - 1) ??
6
??
4
SSC CPO 05/10/2023 (1st Shift)
(a) (1 + ) (b) (1 + x) (c) 1 (d) (1 ) ??
2
- ??
2
Q.4. The LCM of two prime numbers x
and y (x > y) is 533. The value of 4y - x is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 11 (b) 21 (c) 18 (d) 23
Q.5. What is the HCF of the polynomials
( ³ - 8), ( ³ - 6 ² + 12 - 8) and ( ³ - 4 ² + 4 ?? ?? ?? ?? ?? ??
) ? ??
SSC MTS 12/09/2023 (3rd Shift)
(a) ( - 1) (b) ( - 2) (c) ( - 8) (d) ( - 4) ?? ?? ?? ??
Q.6. Find the greatest two digit number
which on dividing 219, 365, 511 leaves
the remainders 3, 5, 7, respectively.
SSC MTS 01/09/2023 (3rd Shift)
(a) 63 (b) 82 (c) 53 (d) 72
Q.7. There is a circular path around a
sports ?eld. Rahul takes 15 minutes to
drive one round of the ?eld, while Anil
takes 18 minutes for the same. Suppose
they both start from the same point and
at the same time, and go in the same
direction. After how many minutes will
they meet again at the starting point?
SSC MTS 11/05/2023 (Afternoon)
(a) 120 (b) 100 (c) 80 (d) 90
Q.8. Let x = 224 and, y = 322. If the
highest common factor of 23x and a × y
is divisible by x and y, then what can be
the possible value of a?
SSC CPO 11/11/2022 (Evening)
(a) 16 (b) 8 (c) 12 (d) 4
Q.9. The highest common factor of 108,
72 and 5a is a. What can be the least
common multiple of 108, 72 and a ?
SSC CPO 10/11/2022 (Afternoon)
(a) 432 (b) 324 (c) 108 (d) 216
Q.10. What is the greatest positive
integer that divides 554, 714 and 213
leaving the remainder 43, 57 and 67,
respectively?
SSC CPO 10/11/2022 (Afternoon)
(a) 95 (b) 71 (c) 83 (d) 73
Q.11. Find the HCF of ( – 1) and 4
315
( – 1). 4
25
SSC MTS 26/07/2022 (Morning)
(a) 1 (b) ( – 1) (c) 1024 (d) 1023 4
25
Q.12. 120 apples, 240 oranges and 150
pears are packed in cartons in such a
way that each carton has the same
number of fruits, each carton contains
only one type of fruit and fruit is left
unpacked. What is the smallest possible
number of cartons needed for the
purpose ?
SSC MTS 08/07/2022 (Evening)
(a) 50 (b) 40 (c) 17 (d) 30
Q.13. 13, a, b, c are four distinct numbers
and the HCF of each pair of numbers (13,
a) : (13, b) : (13, c) is 13, where a, b, c are
each less than 60 and a < b < c. What is
the value of ?
?? + ??
??
SSC CGL 13/04/2021 (Morning)
(a) 3.5 (b) 2 (c) 5 (d) 4.5
Q.14. The sum of two numbers is 1215
and their HCF is 81. If the numbers lie
between 500 and 700, then the sum of
reciprocals of the numbers is
SSC CPO 13/12/2019 (Evening)
(a) (b) (c) (d)
5
1512
5
378
5
702
5
1188
Q.15. If r is the remainder when each of
6454, 7306 and 8797 is divided by the
greatest number d (d > 1), then (d - r) is:
SSC CPO 13/12/2019 (Morning)
(a) 126 (b) 64 (c) 137 (d) 149
Q.16. In ?nding the HCF of two numbers
by division method, the quotients are 1, 8
and 2 respectively, and the last divisor is
105, what is the sum of the numbers?
SSC CPO 11/12/2019 (Evening)
(a) 3570 (b) 3885 (c) 3780 (d) 3675
Q.17. When the smallest number x is
divided by 5, 6, 8, 9 and 12, it gives
remainder 1 in each case. But x is
divisible by 13. What will be the
remainder when x will be divided by 31
SSC MTS 20/08/2019 (Afternoon)
(a) 1 (b) 5 (c) 3 (d) 0
Q.18. The Highest Common Factor and
Lowest Common Multiple of two
numbers p and q are A and B
respectively. IF A + B = p + q, then the
value of is : ??
3
+ ??
3
SSC MTS 09/08/2019 (Evening)
(a) p
3
(b) q
3
(c) p
3
+ q
3
(d) p
3
- q
3
Q.19. What is the largest number that
divides 460, 491 and 553 and leaves the
remainder 26 in each case ?
SSC MTS 06/08/2019 (Afternoon)
(a) 27 (b)35 (c) 33 (d) 31
Q.20. A is the smallest three-digit
number which when divided by 3, 4 and 5
gives remainder 1, 2 and 3 respectively.
What is the sum of the digits of A ?
SSC MTS 05/08/2019 (Afternoon)
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Pinnacle Day: 6th - 7th HCF and LCM
(a) 11 (b) 10 (c) 6 (d) 8
Q.21. The product of two numbers is
6760 and their HCF is 13. How many
such pairs of numbers can be formed?
SSC CPO 16/03/2019 (Evening)
(a) 2 (b) 3 (c) 1 (d) 4
Q.22. An oil merchant has 3 varieties of
oil of volumes 432, 594 and 702
respectively. The number of cans of
equal size that would be required to ?ll
the oil separately is:
SSC CPO 16/03/2019 (Afternoon)
(a) 13, 15, 17 (b) 8, 11, 13
(c) 8, 13, 15 (d) 6, 9, 11
Q.23. Which of the following statements
is true ?
SSC CPO 14/03/2019 (Morning)
(a) LCM of two natural numbers is
divisible by their HCF.
(b) HCF + LCM of two numbers =
Product of the two numbers.
(c) Two prime numbers are co-prime
numbers if their LCM is 1.
(d) HCF of two numbers is the smallest
common divisor of both numbers.
Practice Questions
SSC CPO 2023 Tier - 1
Q.24. The largest number of four digits
that is exactly divisible by 17 and 36 is:
SSC CPO 03/10/2023 (1st Shift)
(a) 8568 (b) 9180 (c) 9792 (d) 7956
Q.25. Two numbers are in the ratio 3 : 4.
The product of their HCF and LCM is
2700. The sum of the numbers is:
SSC CPO 03/10/2023 (1st Shift)
(a) 45 (b) 105 (c) 60 (d) 15
Q.26. Find the greatest number which
when divides 261, 853 and 1221, leaves a
remainder of 5 in each case.
SSC CPO 03/10/2023 (2nd Shift)
(a) 18 (b) 17 (c) 16 (d) 19
Q.27. Find the LCM of , and .
3
2
81
16
9
8
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
111
2
91
2
81
2
101
2
Q.28. Find the greatest possible length
(in metres) that can be used to exactly
measure the lengths 6 m, 5 m 25 cm and
12 m 50 cm.
SSC CPO 03/10/2023 (2nd Shift)
(a) 0.35 m (b) 0.90 m (c) 0.75 m (d) 0.25 m
Q.29. The least number which should be
added to 1351 so that the sum is exactly
divisible by 2, 4, 6 and 8 is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 13 (b) 11 (c) 15 (d) 17
Q.30. Two numbers are in the ratio of 4:
3. The product of their HCF and LCM is
2700. The difference between the
numbers is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 25 (b) 30 (c) 15 (d) 105
Q.31. Let the HCF of m and n be 'a' and
let n = ab, LCM of m and n is given by:
SSC CPO 04/10/2023 (1st Shift)
(a) ab (b) am (c) bm (d) mn
Q.32. The LCM and HCF of two numbers
are 1105 and 5. If the LCM is 17 times
the ?rst number, then ?nd the two
numbers.
SSC CPO 04/10/2023 (2nd Shift)
(a) 55 and 85 (b) 65 and 75
(c) 60 and 80 (d) 65 and 85
Q.33. Which is the smallest natural
number that is exactly divisible by each
of 96, 108 and 144 ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 1728 (b) 864 (c) 1296 (d) 2592
Q.34. The HCF of 3888 and 3969 is:
SSC CPO 04/10/2023 (2nd Shift)
(a) 81 (b) 73 (c) 83 (d) 71
Q.35. A person has three iron bars whose
lengths are 20, 30 and 40 metres,
respectively. He wants to cut pieces of
the same length from each of the three
bars. What is the least number of total
pieces if he cuts without any wastage ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 9 (b) 10 (c) 8 (d) 11
Q.36. If the HCF of 45 and 55 is
expressible in the form of 55 × 5 + 45m,
then what is the value of m ?
SSC CPO 04/10/2023 (3rd Shift)
(a) 5 (b) –6 (c) –5 (d) 6
Q.37. Find the least number which when
divided by 4, 9, 12 and 15, leaves the
remainder 3 in each case.
SSC CPO 04/10/2023 (3rd Shift)
(a) 193 (b) 183 (c) 360 (d) 180
Q.38. Determine the LCM of two
numbers if their HCF is 12 and their ratio
is 13 : 15.
SSC CPO 04/10/2023 (3rd Shift)
(a) 2450 (b) 1780 (c) 1890 (d) 2340
Q.39. What is the smallest perfect
square number which is completely
divisible by 4, 6, 9, 12 and 15?
SSC CPO 04/10/2023 (3rd Shift)
(a) 900 (b) 961 (c) 784 (d) 841
Q.40. Which of the following numbers
leaves the remainder equal to the highest
common factor of 6, 8 and 9, when
divided by 6, 8 and 9?
SSC CPO 04/10/2023 (3rd Shift)
(a) 506 (b) 575 (c) 291 (d) 433
Q.41. Find the least number which is
exactly divisible by 20, 28, 34, 60 and 75.
SSC CPO 05/10/2023 (1st Shift)
(a) 34500 (b) 35900 (c) 35700 (d) 36220
Q.42. Three numbers are in the ratio of 5
: 7 : 9 and their LCM is 34,650. Their HCF
is:
SSC CPO 05/10/2023 (1st Shift)
(a) 110 (b) 315 (c) 99 (d) 55
Q.43. Which is the largest number that
divides each of 1036, 1813 and 3885
without leaving any remainder ?
SSC CPO 05/10/2023 (1st Shift)
(a) 259 (b) 111 (c) 333 (d) 37
Q.44. What is the LCM of 0.15, 0.18 and
0.45 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 0.6 (b) 0.9 (c) 0.81 (d) 0.09
Q.45. What will be the least number
which when doubled will be exactly
divisible by 12, 14, 16 and 18 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 636 (b) 226 (c) 428 (d) 504
Q.46. The LCM of two numbers is ?ve
times their HCF. If the product of the two
numbers is 20480, then ?nd their HCF
and LCM, respectively.
SSC CPO 05/10/2023 (3rd Shift)
(a) 64 and 320 (b) 56 and 280
(c) 48 and 240 (d) 46 and 230
Q.47. If the sum of two numbers is 60
and their HCF and LCM are 5 and 60,
respectively, then the sum of the
reciprocals of the numbers will be:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) (c) (d)
1
4
1
5
1
11
1
6
Q.48. Which of the following is the
greatest four-digit number that is
divisible by 15, 25, 40, and 75?
SSC CPO 05/10/2023 (3rd Shift)
(a) 9000 (b) 9600 (c) 9500 (d) 9200
Q.49. The least number of ?ve digits
which is exactly divisible by 9, 12, 15, 25
and 27 is:
SSC CPO 05/10/2023 (3rd Shift)
(a) 10250 (b) 10800 (c) 10600 (d) 10700
SSC MTS 2023 Tier - 1
Q.50. Three measuring tapes are 64 cm
72 cm and 96 cm, respectively. What is
the least length that can be measured by
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 29
Pinnacle Day: 6th - 7th HCF and LCM
any of the tapes exactly (in cm)?
SSC MTS 04/09/2023 (1st Shift)
(a) 575 (b) 570 (c) 576 (d) 525
Q.51. What is the largest number that
divides 627, 15630 and 3128 and leaves
remainders of 2, 5 and 3, respectively?
SSC MTS 04/09/2023 (3rd Shift)
(a) 775 (b) 650 (c) 625 (d) 1225
Q.52. The HCF of three numbers is 57. If
they are in the ratio of 4 : 5 : 6, then ?nd
the numbers.
SSC MTS 05/09/2023 (2nd Shift)
(a) 228, 285, 342 (b) 236, 295, 354
(c) 240, 300, 360 (d) 232, 290, 348
Q.53. Three numbers are in the ratio 3 : 5
: 7 If their LCM is 2625, ?nd their HCF.
SSC MTS 05/09/2023 (3rd Shift)
(a) 100 (b) 125 (c) 50 (d) 25
Q.54. Two tankers contain 850 litres and
680 litres of oil. Find the maximum
capacity of a container which can
measure the oil of both tankers when
used, an exact number of times.
SSC MTS 06/09/2023 (2nd Shift)
(a) 425 litres (b) 680 litres
(c) 340 litres (d) 170 litres
Q.55. The largest three-digit number
which gives the same remainder 2 when
divided by 3, 5 and 9 is ________.
SSC MTS 08/09/2023 (1st Shift)
(a) 980 (b) 992 (c) 995 (d) 990
Q.56. The LCM of , , , is:
1
6
5
27
4
15
8
3
SSC MTS 08/09/2023 (3rd Shift)
(a) (b) (c) (d)
40
3
3
28
5
28
40
5
Q.57. Sanjay wants to exactly measure
the lengths 7000 mm, 3850 mm and
12950 mm using a single measuring
tape. What will be the greatest possible
length (in cm) of the measuring tape?
SSC MTS 11/09/2023 (2nd Shift)
(a) 40 (b) 45 (c) 35 (d) 30
Q.58. Radha, Pratima and Reena begin to
jog around a circular path and they
complete their revolutions in 50 seconds,
75 seconds and 100 seconds,
respectively. After how much time (in
minutes) will they meet together at the
starting point for the ?rst time?
SSC MTS 11/09/2023 (3rd Shift)
(a) 6 (b) 5 (c) 4 (d) 3
Q.59. The HCF of two numbers is 12.
Which of the following can never be their
LCM ?
SSC MTS 12/09/2023 (1st Shift)
(a) 24 (b) 74 (c) 48 (d) 60
Q.60. The product of two numbers is
4107. If the HCF of these numbers is 37,
then the greater number is:
SSC MTS 12/09/2023 (2nd Shift)
(a) 111 (b) 74 (c) 185 (d) 37
Q.61. Two numbers are in the ratio of
13 : 15. If their HCF is 3, then what is the
difference between these two numbers ?
SSC MTS 13/09/2023 (1st Shift)
(a) 4 (b) 2 (c) 8 (d) 6
Q.62. The LCM of the two numbers is
1440 and their HCF is 32. If the ?rst
number is 288, then ?nd the second
number.
SSC MTS 13/09/2023 (2nd Shift)
(a) 150 (b) 145 (c) 180 (d) 160
Q.63. What is the greatest three-digit
number which when divided by 4, 8 and
9, leaves no remainder?
SSC MTS 13/09/2023 (3rd Shift)
(a) 924 (b) 963 (c) 932 (d) 936
Q.64. Find the smallest three digit
number which when added to 5 is exactly
divisible by 12, 14 and 18.
SSC MTS 14/09/2023 (2nd Shift)
(a) 240 (b) 225 (c) 245 (d) 247
Q.65. The ratio of two numbers is 3 : 4. If
the sum of these two numbers is 70, then
what is their LCM ?
SSC MTS 14/09/2023 (3rd Shift)
(a) 120 (b) 96 (c) 108 (d) 84
SSC CGL 2023 Tier - 1
Q.66. Find the least number divisible by
2, 3, 5, 6, 9 and 18, which is a perfect
square.
SSC CGL 24/07/2023 (2nd shift)
(a) 900 (b) 400 (c) 144 (d) 3600
SSC Selection Post (Phase - XI)
Q.67. The smallest six-digit number
which is divisible by 12, 16, 24, 28 is:
Matriculation Level 30/06/2023 (Shift - 3)
(a) 100128 (b) 100190
(c) 100180 (d) 100160
SSC CHSL 2022 Tier - 2
Q.68. The HCF of two numbers is
one-twentieth of their LCM. If one of the
numbers is 96 and the difference of the
LCM and the HCF is 456, then what is the
other number ?
SSC CHSL Tier II (26/06/2023)
(a) 48 (b) 120 (c) 144 (d) 72
SSC MTS 2022 Tier - 1
Q.69. What is the HCF of two prime
numbers X and Y ?
SSC MTS 02/05/2023 (Morning)
(a) 1 (b) 2 (c) Y (d) X
Q.70. Two numbers are in the ratio 11 : 6.
If their HCF is 4, ?nd the smallest
number.
SSC MTS 02/05/2023 (Afternoon)
(a) 30 (b) 18 (c) 24 (d) 12
Q.71. Three metal rods of length 77 cm,
110 cm and 121 cm are to be cut into
parts of equal length. Each part must be
as long as possible. What is the maximum
number of pieces that can be cut?
SSC MTS 03/05/2023 (Morning)
(a) 18 (b) 28 (c) 11 (d) 21
Q.72. What is that least number that
should be added to 478 so that the
number obtained becomes exactly
divisible by 5, 6 and 12 ?
SSC MTS 04/05/2023 (Morning)
(a) 62 (b) 12 (c) 2 (d) 52
Q.73. The product of two co-prime
numbers is 483. Find the Least Common
Multiple of both numbers.
SSC MTS 08/05/2023 (Afternoon)
(a) 483 (b) 21 (c) 1 (d) 23
Q.74. What is that greatest four-digit
number which when divided by 3 and 4
leaves remainder 2 in each case?
SSC MTS 09/05/2023 (Morning)
(a) 9994 (b) 9996 (c) 9998 (d) 9995
Q.75. What is that least perfect square
which is exactly divisible by each of 14,
84 and 28 ?
SSC MTS 10/05/2023 (Afternoon)
(a) 7056 (b) 3528 (c) 1764 (d) 441
Q.76. Find the LCM of reciprocals of 22
and 16.
SSC MTS 11/05/2023 (Evening)
(a) (b) (c) (d)
1
2
1
3
1
6
1
4
Q.77. The greatest number which when
divides 456 and 553 leaves the
remainder as 6 and 3 respectively is:
SSC MTS 12/05/2023 (Evening)
(a) 5 (b) 50 (c) 10 (d) 30
Q.78. The least number which when
divided by 14, 18 and 36 and leaves 1 as
remainder in each case is:
SSC MTS 16/05/2023 (Morning)
(a) 250 (b) 252 (c) 253 (d) 251
Q.79. Three numbers are in the ratio 2 : 3
: 5 and their LCM is 60. What is the HCF
of the numbers?
SSC MTS 16/05/2023 (Evening)
(a) 2 (b) 4 (c) 10 (d) 6
Q.80. Five bells commence tolling
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Page 5
Pinnacle Day: 6th - 7th HCF and LCM
HCF and LCM
LCM (Least common multiple) of two or
more given numbers is the least number
that is exactly divisible by each of them.
HCF (Highest common factor) of two or
more numbers is the greatest number
that divides each of them exactly. HCF is
also known as the ‘Highest common
Divisor’ (HCD) and the Greatest Common
Measure (GCM).
The concept of multiples and factors
If X , Y , and Z are three natural ?
numbers and X × Y = Z , then
(i). X and Y are called the factors of
Z.
(ii). Z is said to be divisible by X and
Y.
(iii). Z is said to be a multiple of X
and Y.
Example: The set of positive Integers
which are factors of 18 is (1, 2, 3, 6, 9,
18).
Basic Concepts of H.C.F. and LCM
Method of finding H.C.F.
To ?nd the HCF of the given numbers
1. Break the given numbers into their
prime factors.
2. The HCF will be the product of all the
prime factors common to all the
numbers.
Let us learn the process of ?nding HCF
with the help of some solved examples.
Example:-
Find the HCF of 96, 36 and 18 ?
Solution:-
96 = 2 × 3 × 2 × 2 × 2 × 2
36 = 2 × 3 × 2 × 3
18 = 2 × 3 × 3
Therefore, the HCF of 96, 36 and 18 is
the product of the highest number of
common factors in the given numbers
i.e., 2 × 3 = 6. In other words, 6 is the
largest possible integer, which can divide
96, 36 and 18 without leaving any
remainder.
Example:- Find the H.C.F. of 42 and 70 ?
Solution:- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, H.C.F. of 42 and 70 = 2 × 7
= 14.
HCF by Division Method
Example:- HCF of 24, 48, 72, and 100.
Solution:- To start the division method
select the smallest two numbers./
HCF of 24 and 48 = 24
HCF of 24, 48 and 72 = 24
HCF of 24, 48, 72, and 100 = 4.
Example:- HCF of 1785, 1995, 3381.
Solution:-
HCF of 1785 and 1995 = 105
HCF of 1785, 1995 and 3381 = 21
NOTE :-
(i). HCF of two prime numbers is
always 1.
(ii). HCF of co-prime numbers is
always 1.
Method of finding L.C.M
The Least Common Multiple of two or
more numbers is the smallest number
which is exactly divisible by all of them.
In other words, it is the product of the
highest powers of all the prime factors of
the given numbers.
To ?nd the LCM of given numbers:
1. Break the given numbers into
their prime factors.
2. The LCM will be the product of
the highest power of all the
factors that occur in the given
numbers.
Let us take some solved examples.
Example:- Find the LCM of 96, 36 and 18.
Solution : 96 = 2 × 2 × 2 × 2 × 2 × 3
= × ; 36 = 2 × 2 × 3 × 3 = × 2
5
3
1
2
2
3
2
18 = 2 × 3 × 3 = × 2
1
3
2
Therefore, LCM of 96, 36 and 18 is the
product of the highest powers of all the
prime factors, i.e. 2
5
× 3
2
= 32 × 9 = 288
That is, 288 is the smallest integer which
is divisible by 96, 36 and 18 without
leaving any remainder.
Example:- Find the LCM of 42 and 70
Solution :- 42 = 3 × 2 × 7
70 = 5 × 2 × 7
Hence, LCM is 2 × 3 × 5 × 7 = 210.
Example:- LCM of 6, 12, 8 ?
Solution :-
LCM = 2 × 2 × 3 × 2 = 24
HCF of 6, 12, 18
Firstly ?nd out the factors of 6, 12, 18
and then multiply the common factors.
6 = , 12 = , 18 = 2 × 3 2 × 2 × 3 2 × 3 × 3
HCF = 2 × 3 = 6
Try ?nding HCF and LCM of 3, 6, 9, 12
yourself. HCF of 3, 6, 9, 12 can also be
found by Division method. It is useful
when the numbers are bigger.
NOTE :-
(i). HCF of A, B and C is the highest
divisor which can exactly divide
A, B, and C.
(ii). LCM of A, B and C is the lowest
dividend which is exactly
divisible by A, B, and C.
There is one very important relationship,
given below, between two numbers and
their HCF and LCM. Many problems have
appeared in various competitive exams
based on this relationship.
Important Concepts:-
(1).
LCM × HCF = 1st number × 2nd number
Example:- For numbers 8 and 12,
LCM = 24 and HCF = 4
Now, LCM HCF = 24 4 = 96 × ×
also, 8 12 = 96 ×
(2). HCF of some numbers is always a
factor of LCM of the numbers.
(3).
LCM of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
(4).
HCF of Fraction =
?????? ???? ??????????????????
?????? ???? ??????????????????????
Example: LCM and HCF of , and
1
2
2
3
3
4
Solution:- LCM =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
6
1
HCF =
?????? ???? ??????????????????
?????? ???? ??????????????????????
= =
?????? ???? 1 , 2 , 3
?????? ???? 2 , 3 , 4
1
12
(5) Co- Prime numbers :
If the HCF of two numbers is 1 then they
are called co-prime numbers.
(6) = ;
??????
??????
??????????????
where LCM and HCF are of two numbers
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Pinnacle Day: 6th - 7th HCF and LCM
. If we ?nd two co-prime ??
1
?????? ??
2
factors, , of the Product ??
1
?????? ??
2
as obtained above.
(7)
Example:-
(8).
If 1st number = and 2nd number = ??
1
??
2
HCF of = Hx ??
1
HCF of = Hy ??
2
And HCF of and = H ??
1
??
2
So ,
Difference between and = Hx - Hy ??
1
??
2
= H(x - y)
Example :-
= 24 and = 36 ??
1
??
2
Note :- HCF is always either the
difference of two numbers or factors of
difference of two numbers.
(9). When the second divisor is a factor
of the ?rst divisor, then the second
remainder is obtained by dividing the ?rst
remainder by the second divisor.
Example :- When 29 is divided by 8, the
remainder obtained is 5, then what will
be the remainder when the same number
is divided by 4 ?
Solution:- Here, the second divisor that is
4 is a factor of the ?rst divisor that is 8.
So, on dividing the ?rst remainder that is
5 by the second divisor that is 4, we get
our second remainder which is 1. So, the
required answer is 1.
Variety Questions
Q.1. The LCM of x² - 8x + 15 and x² - 5x + 6
is:
SSC CPO 05/10/2023 (2nd Shift)
(a) (x + 5)(x + 2)(x + 3)
(b) (x - 5)(x - 2)(x - 3)
(c) (x + 5)(x - 2)(x - 3)
(d) (x - 2)(x - 3)
2
(x - 5)
Q.2. The greatest possible length that
can be used to measure exactly the
lengths of 3 m 15 cm, 5 m, and 6 m 85
cm is:
SSC CPO 05/10/2023 (2nd Shift)
(a) 11 cm (b) 7 cm (c) 9 cm (d) 5 cm
Q.3. What is the HCF of ( + 1) and ( - 1) ??
6
??
4
SSC CPO 05/10/2023 (1st Shift)
(a) (1 + ) (b) (1 + x) (c) 1 (d) (1 ) ??
2
- ??
2
Q.4. The LCM of two prime numbers x
and y (x > y) is 533. The value of 4y - x is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 11 (b) 21 (c) 18 (d) 23
Q.5. What is the HCF of the polynomials
( ³ - 8), ( ³ - 6 ² + 12 - 8) and ( ³ - 4 ² + 4 ?? ?? ?? ?? ?? ??
) ? ??
SSC MTS 12/09/2023 (3rd Shift)
(a) ( - 1) (b) ( - 2) (c) ( - 8) (d) ( - 4) ?? ?? ?? ??
Q.6. Find the greatest two digit number
which on dividing 219, 365, 511 leaves
the remainders 3, 5, 7, respectively.
SSC MTS 01/09/2023 (3rd Shift)
(a) 63 (b) 82 (c) 53 (d) 72
Q.7. There is a circular path around a
sports ?eld. Rahul takes 15 minutes to
drive one round of the ?eld, while Anil
takes 18 minutes for the same. Suppose
they both start from the same point and
at the same time, and go in the same
direction. After how many minutes will
they meet again at the starting point?
SSC MTS 11/05/2023 (Afternoon)
(a) 120 (b) 100 (c) 80 (d) 90
Q.8. Let x = 224 and, y = 322. If the
highest common factor of 23x and a × y
is divisible by x and y, then what can be
the possible value of a?
SSC CPO 11/11/2022 (Evening)
(a) 16 (b) 8 (c) 12 (d) 4
Q.9. The highest common factor of 108,
72 and 5a is a. What can be the least
common multiple of 108, 72 and a ?
SSC CPO 10/11/2022 (Afternoon)
(a) 432 (b) 324 (c) 108 (d) 216
Q.10. What is the greatest positive
integer that divides 554, 714 and 213
leaving the remainder 43, 57 and 67,
respectively?
SSC CPO 10/11/2022 (Afternoon)
(a) 95 (b) 71 (c) 83 (d) 73
Q.11. Find the HCF of ( – 1) and 4
315
( – 1). 4
25
SSC MTS 26/07/2022 (Morning)
(a) 1 (b) ( – 1) (c) 1024 (d) 1023 4
25
Q.12. 120 apples, 240 oranges and 150
pears are packed in cartons in such a
way that each carton has the same
number of fruits, each carton contains
only one type of fruit and fruit is left
unpacked. What is the smallest possible
number of cartons needed for the
purpose ?
SSC MTS 08/07/2022 (Evening)
(a) 50 (b) 40 (c) 17 (d) 30
Q.13. 13, a, b, c are four distinct numbers
and the HCF of each pair of numbers (13,
a) : (13, b) : (13, c) is 13, where a, b, c are
each less than 60 and a < b < c. What is
the value of ?
?? + ??
??
SSC CGL 13/04/2021 (Morning)
(a) 3.5 (b) 2 (c) 5 (d) 4.5
Q.14. The sum of two numbers is 1215
and their HCF is 81. If the numbers lie
between 500 and 700, then the sum of
reciprocals of the numbers is
SSC CPO 13/12/2019 (Evening)
(a) (b) (c) (d)
5
1512
5
378
5
702
5
1188
Q.15. If r is the remainder when each of
6454, 7306 and 8797 is divided by the
greatest number d (d > 1), then (d - r) is:
SSC CPO 13/12/2019 (Morning)
(a) 126 (b) 64 (c) 137 (d) 149
Q.16. In ?nding the HCF of two numbers
by division method, the quotients are 1, 8
and 2 respectively, and the last divisor is
105, what is the sum of the numbers?
SSC CPO 11/12/2019 (Evening)
(a) 3570 (b) 3885 (c) 3780 (d) 3675
Q.17. When the smallest number x is
divided by 5, 6, 8, 9 and 12, it gives
remainder 1 in each case. But x is
divisible by 13. What will be the
remainder when x will be divided by 31
SSC MTS 20/08/2019 (Afternoon)
(a) 1 (b) 5 (c) 3 (d) 0
Q.18. The Highest Common Factor and
Lowest Common Multiple of two
numbers p and q are A and B
respectively. IF A + B = p + q, then the
value of is : ??
3
+ ??
3
SSC MTS 09/08/2019 (Evening)
(a) p
3
(b) q
3
(c) p
3
+ q
3
(d) p
3
- q
3
Q.19. What is the largest number that
divides 460, 491 and 553 and leaves the
remainder 26 in each case ?
SSC MTS 06/08/2019 (Afternoon)
(a) 27 (b)35 (c) 33 (d) 31
Q.20. A is the smallest three-digit
number which when divided by 3, 4 and 5
gives remainder 1, 2 and 3 respectively.
What is the sum of the digits of A ?
SSC MTS 05/08/2019 (Afternoon)
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Pinnacle Day: 6th - 7th HCF and LCM
(a) 11 (b) 10 (c) 6 (d) 8
Q.21. The product of two numbers is
6760 and their HCF is 13. How many
such pairs of numbers can be formed?
SSC CPO 16/03/2019 (Evening)
(a) 2 (b) 3 (c) 1 (d) 4
Q.22. An oil merchant has 3 varieties of
oil of volumes 432, 594 and 702
respectively. The number of cans of
equal size that would be required to ?ll
the oil separately is:
SSC CPO 16/03/2019 (Afternoon)
(a) 13, 15, 17 (b) 8, 11, 13
(c) 8, 13, 15 (d) 6, 9, 11
Q.23. Which of the following statements
is true ?
SSC CPO 14/03/2019 (Morning)
(a) LCM of two natural numbers is
divisible by their HCF.
(b) HCF + LCM of two numbers =
Product of the two numbers.
(c) Two prime numbers are co-prime
numbers if their LCM is 1.
(d) HCF of two numbers is the smallest
common divisor of both numbers.
Practice Questions
SSC CPO 2023 Tier - 1
Q.24. The largest number of four digits
that is exactly divisible by 17 and 36 is:
SSC CPO 03/10/2023 (1st Shift)
(a) 8568 (b) 9180 (c) 9792 (d) 7956
Q.25. Two numbers are in the ratio 3 : 4.
The product of their HCF and LCM is
2700. The sum of the numbers is:
SSC CPO 03/10/2023 (1st Shift)
(a) 45 (b) 105 (c) 60 (d) 15
Q.26. Find the greatest number which
when divides 261, 853 and 1221, leaves a
remainder of 5 in each case.
SSC CPO 03/10/2023 (2nd Shift)
(a) 18 (b) 17 (c) 16 (d) 19
Q.27. Find the LCM of , and .
3
2
81
16
9
8
SSC CPO 03/10/2023 (2nd Shift)
(a) (b) (c) (d)
111
2
91
2
81
2
101
2
Q.28. Find the greatest possible length
(in metres) that can be used to exactly
measure the lengths 6 m, 5 m 25 cm and
12 m 50 cm.
SSC CPO 03/10/2023 (2nd Shift)
(a) 0.35 m (b) 0.90 m (c) 0.75 m (d) 0.25 m
Q.29. The least number which should be
added to 1351 so that the sum is exactly
divisible by 2, 4, 6 and 8 is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 13 (b) 11 (c) 15 (d) 17
Q.30. Two numbers are in the ratio of 4:
3. The product of their HCF and LCM is
2700. The difference between the
numbers is:
SSC CPO 03/10/2023 (3rd Shift)
(a) 25 (b) 30 (c) 15 (d) 105
Q.31. Let the HCF of m and n be 'a' and
let n = ab, LCM of m and n is given by:
SSC CPO 04/10/2023 (1st Shift)
(a) ab (b) am (c) bm (d) mn
Q.32. The LCM and HCF of two numbers
are 1105 and 5. If the LCM is 17 times
the ?rst number, then ?nd the two
numbers.
SSC CPO 04/10/2023 (2nd Shift)
(a) 55 and 85 (b) 65 and 75
(c) 60 and 80 (d) 65 and 85
Q.33. Which is the smallest natural
number that is exactly divisible by each
of 96, 108 and 144 ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 1728 (b) 864 (c) 1296 (d) 2592
Q.34. The HCF of 3888 and 3969 is:
SSC CPO 04/10/2023 (2nd Shift)
(a) 81 (b) 73 (c) 83 (d) 71
Q.35. A person has three iron bars whose
lengths are 20, 30 and 40 metres,
respectively. He wants to cut pieces of
the same length from each of the three
bars. What is the least number of total
pieces if he cuts without any wastage ?
SSC CPO 04/10/2023 (2nd Shift)
(a) 9 (b) 10 (c) 8 (d) 11
Q.36. If the HCF of 45 and 55 is
expressible in the form of 55 × 5 + 45m,
then what is the value of m ?
SSC CPO 04/10/2023 (3rd Shift)
(a) 5 (b) –6 (c) –5 (d) 6
Q.37. Find the least number which when
divided by 4, 9, 12 and 15, leaves the
remainder 3 in each case.
SSC CPO 04/10/2023 (3rd Shift)
(a) 193 (b) 183 (c) 360 (d) 180
Q.38. Determine the LCM of two
numbers if their HCF is 12 and their ratio
is 13 : 15.
SSC CPO 04/10/2023 (3rd Shift)
(a) 2450 (b) 1780 (c) 1890 (d) 2340
Q.39. What is the smallest perfect
square number which is completely
divisible by 4, 6, 9, 12 and 15?
SSC CPO 04/10/2023 (3rd Shift)
(a) 900 (b) 961 (c) 784 (d) 841
Q.40. Which of the following numbers
leaves the remainder equal to the highest
common factor of 6, 8 and 9, when
divided by 6, 8 and 9?
SSC CPO 04/10/2023 (3rd Shift)
(a) 506 (b) 575 (c) 291 (d) 433
Q.41. Find the least number which is
exactly divisible by 20, 28, 34, 60 and 75.
SSC CPO 05/10/2023 (1st Shift)
(a) 34500 (b) 35900 (c) 35700 (d) 36220
Q.42. Three numbers are in the ratio of 5
: 7 : 9 and their LCM is 34,650. Their HCF
is:
SSC CPO 05/10/2023 (1st Shift)
(a) 110 (b) 315 (c) 99 (d) 55
Q.43. Which is the largest number that
divides each of 1036, 1813 and 3885
without leaving any remainder ?
SSC CPO 05/10/2023 (1st Shift)
(a) 259 (b) 111 (c) 333 (d) 37
Q.44. What is the LCM of 0.15, 0.18 and
0.45 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 0.6 (b) 0.9 (c) 0.81 (d) 0.09
Q.45. What will be the least number
which when doubled will be exactly
divisible by 12, 14, 16 and 18 ?
SSC CPO 05/10/2023 (2nd Shift)
(a) 636 (b) 226 (c) 428 (d) 504
Q.46. The LCM of two numbers is ?ve
times their HCF. If the product of the two
numbers is 20480, then ?nd their HCF
and LCM, respectively.
SSC CPO 05/10/2023 (3rd Shift)
(a) 64 and 320 (b) 56 and 280
(c) 48 and 240 (d) 46 and 230
Q.47. If the sum of two numbers is 60
and their HCF and LCM are 5 and 60,
respectively, then the sum of the
reciprocals of the numbers will be:
SSC CPO 05/10/2023 (3rd Shift)
(a) (b) (c) (d)
1
4
1
5
1
11
1
6
Q.48. Which of the following is the
greatest four-digit number that is
divisible by 15, 25, 40, and 75?
SSC CPO 05/10/2023 (3rd Shift)
(a) 9000 (b) 9600 (c) 9500 (d) 9200
Q.49. The least number of ?ve digits
which is exactly divisible by 9, 12, 15, 25
and 27 is:
SSC CPO 05/10/2023 (3rd Shift)
(a) 10250 (b) 10800 (c) 10600 (d) 10700
SSC MTS 2023 Tier - 1
Q.50. Three measuring tapes are 64 cm
72 cm and 96 cm, respectively. What is
the least length that can be measured by
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Pinnacle Day: 6th - 7th HCF and LCM
any of the tapes exactly (in cm)?
SSC MTS 04/09/2023 (1st Shift)
(a) 575 (b) 570 (c) 576 (d) 525
Q.51. What is the largest number that
divides 627, 15630 and 3128 and leaves
remainders of 2, 5 and 3, respectively?
SSC MTS 04/09/2023 (3rd Shift)
(a) 775 (b) 650 (c) 625 (d) 1225
Q.52. The HCF of three numbers is 57. If
they are in the ratio of 4 : 5 : 6, then ?nd
the numbers.
SSC MTS 05/09/2023 (2nd Shift)
(a) 228, 285, 342 (b) 236, 295, 354
(c) 240, 300, 360 (d) 232, 290, 348
Q.53. Three numbers are in the ratio 3 : 5
: 7 If their LCM is 2625, ?nd their HCF.
SSC MTS 05/09/2023 (3rd Shift)
(a) 100 (b) 125 (c) 50 (d) 25
Q.54. Two tankers contain 850 litres and
680 litres of oil. Find the maximum
capacity of a container which can
measure the oil of both tankers when
used, an exact number of times.
SSC MTS 06/09/2023 (2nd Shift)
(a) 425 litres (b) 680 litres
(c) 340 litres (d) 170 litres
Q.55. The largest three-digit number
which gives the same remainder 2 when
divided by 3, 5 and 9 is ________.
SSC MTS 08/09/2023 (1st Shift)
(a) 980 (b) 992 (c) 995 (d) 990
Q.56. The LCM of , , , is:
1
6
5
27
4
15
8
3
SSC MTS 08/09/2023 (3rd Shift)
(a) (b) (c) (d)
40
3
3
28
5
28
40
5
Q.57. Sanjay wants to exactly measure
the lengths 7000 mm, 3850 mm and
12950 mm using a single measuring
tape. What will be the greatest possible
length (in cm) of the measuring tape?
SSC MTS 11/09/2023 (2nd Shift)
(a) 40 (b) 45 (c) 35 (d) 30
Q.58. Radha, Pratima and Reena begin to
jog around a circular path and they
complete their revolutions in 50 seconds,
75 seconds and 100 seconds,
respectively. After how much time (in
minutes) will they meet together at the
starting point for the ?rst time?
SSC MTS 11/09/2023 (3rd Shift)
(a) 6 (b) 5 (c) 4 (d) 3
Q.59. The HCF of two numbers is 12.
Which of the following can never be their
LCM ?
SSC MTS 12/09/2023 (1st Shift)
(a) 24 (b) 74 (c) 48 (d) 60
Q.60. The product of two numbers is
4107. If the HCF of these numbers is 37,
then the greater number is:
SSC MTS 12/09/2023 (2nd Shift)
(a) 111 (b) 74 (c) 185 (d) 37
Q.61. Two numbers are in the ratio of
13 : 15. If their HCF is 3, then what is the
difference between these two numbers ?
SSC MTS 13/09/2023 (1st Shift)
(a) 4 (b) 2 (c) 8 (d) 6
Q.62. The LCM of the two numbers is
1440 and their HCF is 32. If the ?rst
number is 288, then ?nd the second
number.
SSC MTS 13/09/2023 (2nd Shift)
(a) 150 (b) 145 (c) 180 (d) 160
Q.63. What is the greatest three-digit
number which when divided by 4, 8 and
9, leaves no remainder?
SSC MTS 13/09/2023 (3rd Shift)
(a) 924 (b) 963 (c) 932 (d) 936
Q.64. Find the smallest three digit
number which when added to 5 is exactly
divisible by 12, 14 and 18.
SSC MTS 14/09/2023 (2nd Shift)
(a) 240 (b) 225 (c) 245 (d) 247
Q.65. The ratio of two numbers is 3 : 4. If
the sum of these two numbers is 70, then
what is their LCM ?
SSC MTS 14/09/2023 (3rd Shift)
(a) 120 (b) 96 (c) 108 (d) 84
SSC CGL 2023 Tier - 1
Q.66. Find the least number divisible by
2, 3, 5, 6, 9 and 18, which is a perfect
square.
SSC CGL 24/07/2023 (2nd shift)
(a) 900 (b) 400 (c) 144 (d) 3600
SSC Selection Post (Phase - XI)
Q.67. The smallest six-digit number
which is divisible by 12, 16, 24, 28 is:
Matriculation Level 30/06/2023 (Shift - 3)
(a) 100128 (b) 100190
(c) 100180 (d) 100160
SSC CHSL 2022 Tier - 2
Q.68. The HCF of two numbers is
one-twentieth of their LCM. If one of the
numbers is 96 and the difference of the
LCM and the HCF is 456, then what is the
other number ?
SSC CHSL Tier II (26/06/2023)
(a) 48 (b) 120 (c) 144 (d) 72
SSC MTS 2022 Tier - 1
Q.69. What is the HCF of two prime
numbers X and Y ?
SSC MTS 02/05/2023 (Morning)
(a) 1 (b) 2 (c) Y (d) X
Q.70. Two numbers are in the ratio 11 : 6.
If their HCF is 4, ?nd the smallest
number.
SSC MTS 02/05/2023 (Afternoon)
(a) 30 (b) 18 (c) 24 (d) 12
Q.71. Three metal rods of length 77 cm,
110 cm and 121 cm are to be cut into
parts of equal length. Each part must be
as long as possible. What is the maximum
number of pieces that can be cut?
SSC MTS 03/05/2023 (Morning)
(a) 18 (b) 28 (c) 11 (d) 21
Q.72. What is that least number that
should be added to 478 so that the
number obtained becomes exactly
divisible by 5, 6 and 12 ?
SSC MTS 04/05/2023 (Morning)
(a) 62 (b) 12 (c) 2 (d) 52
Q.73. The product of two co-prime
numbers is 483. Find the Least Common
Multiple of both numbers.
SSC MTS 08/05/2023 (Afternoon)
(a) 483 (b) 21 (c) 1 (d) 23
Q.74. What is that greatest four-digit
number which when divided by 3 and 4
leaves remainder 2 in each case?
SSC MTS 09/05/2023 (Morning)
(a) 9994 (b) 9996 (c) 9998 (d) 9995
Q.75. What is that least perfect square
which is exactly divisible by each of 14,
84 and 28 ?
SSC MTS 10/05/2023 (Afternoon)
(a) 7056 (b) 3528 (c) 1764 (d) 441
Q.76. Find the LCM of reciprocals of 22
and 16.
SSC MTS 11/05/2023 (Evening)
(a) (b) (c) (d)
1
2
1
3
1
6
1
4
Q.77. The greatest number which when
divides 456 and 553 leaves the
remainder as 6 and 3 respectively is:
SSC MTS 12/05/2023 (Evening)
(a) 5 (b) 50 (c) 10 (d) 30
Q.78. The least number which when
divided by 14, 18 and 36 and leaves 1 as
remainder in each case is:
SSC MTS 16/05/2023 (Morning)
(a) 250 (b) 252 (c) 253 (d) 251
Q.79. Three numbers are in the ratio 2 : 3
: 5 and their LCM is 60. What is the HCF
of the numbers?
SSC MTS 16/05/2023 (Evening)
(a) 2 (b) 4 (c) 10 (d) 6
Q.80. Five bells commence tolling
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Pinnacle Day: 6th - 7th HCF and LCM
together and toll at intervals of 5, 10, 15,
20 and 25 seconds respectively. In 1 hour
20 minutes, how many times do they toll
together?
SSC MTS 17/05/2023 (Afternoon)
(a) 20 times (b) 15 times
(c) 14 times (d) 17 times
Q.81. What will be the ratio of the Least
Common Multiple and Highest Common
Factor of 12, 24 and 36?
SSC MTS 13/06/2023 (Afternoon)
(a) 3 : 1 (b) 4 : 1 (c) 6 : 1 (d) 2 : 1
Q.82. What is the least number of
students in a class such that they can
make rows of 10, 15 and 20 and also the
total number of students is a perfect
square?
SSC MTS 15/06/2023 (Morning)
(a) 6400 (b) 1600 (c) 900 (d) 1800
Q.83. The sum of the two numbers is 45.
Their HCF and LCM are 5 and 70
respectively. Find the sum of the
reciprocal of the two numbers.
SSC MTS 16/06/2023 (Afternoon)
(a) (b) (c) (d)
9
70
3
140
4
35
3
16
Q.84. What is the HCF of 513 and 1017 ?
SSC MTS 16/06/2023 (Evening)
(a) 15 (b) 8 (c) 9 (d) 13
Q.85. The Least Common Multiple of
three different numbers is 40. Which of
the following cannot be their Highest
Common Factor ?
SSC MTS 19/06/2023 (Morning)
(a) 20 (b) 25 (c) 5 (d) 10
Q.86. What is the least number which
when divided by 8, 10, 12 and 18 leaves a
remainder 5 in each case?
SSC MTS 20/06/2023 (Morning)
(a) 397 (b) 365 (c) 371 (d) 408
Q.87. What is the smallest perfect square
which is divisible by both 8 and 12 ?
SSC MTS 20/06/2023 (Afternoon)
(a) 100 (b) 144 (c) 121 (d) 196
Q.88. The length, breadth, height of a
room is 6 m, 4 m 80 cm and 3 m 60 cm
respectively. Find the longest tape which
can measure the dimensions of the room
exactly.
SSC MTS 20/06/2023 (Evening)
(a) 1 m 40 cm (b) 1m 20 cm
(c) 1 m 80 cm (d) 1 m 50 cm
SSC CGL 2022 Tier - 2
Q.89. The product of the two numbers is
1500 and their HCF is 10. The number of
such possible pairs is/are:
SSC CGL Tier II (02/03/2023)
(a) 1 (b) 3 (c) 4 (d) 2
Q.90. The LCM of and ?? ² - 8 ?? + 15
is: ?? ² - 5 ?? + 6
SSC CGL Tier II (03/03/2023)
(a) ( 2)( 3)( 5) ?? - ?? - ?? -
(b) ( 6)² ( + 1) ( - 3) ?? - ?? ??
(c) ( 6) ( + 1)( 3) ?? - ?? ?? -
(d) ( + 6)( + 1)( 3) ?? ?? ?? -
Q.91. If the highest common factor
(HCF) of x and y is 15, then the HCF of
36 - 81 and 81 - 9 is divisible by ??
2
??
2
??
2
??
2
SSC CGL Tier II (06/03/2023)
(a) 135 (b) 120 (c) 180 (d) 90
SSC CGL 2022 Tier - 1
Q.92. What will be the least number
which when doubled will be exactly
divisible by 15, 18, 25 and 32 ?
SSC CGL 02/12/2022 (2nd Shift)
(a) 3600 (b) 7200 (c) 6400 (d) 3200
Q.93. Calculate the HCF of , &
12
5
14
15
16
17
SSC CGL 03/12/2022 (1st Shift)
(a) (b) (c) (d)
4
255
3
255
2
255
1
255
Q.94. The LCM of two numbers is 120
and the numbers are in the ratio 3 : 8.
The sum of the numbers will be:
SSC CGL 03/12/2022 (3rd Shift)
(a) 48 (b) 55 (c) 45 (d) 60
Q.95. Choose the correct statement from
the following.
SSC CGL 06/12/2022 (2nd Shift)
(a) HCF is the least common multiple of
the given numbers.
(b) HCF of two or more numbers is the
highest number which perfectly
divides all the given numbers.
(c) HCF is also called the least common
divisor.
(d) In prime factorisation method of HCF,
the multiples of all the given numbers
are listed.
SSC CPO 2022 Tier - 1
Q.96. The least common multiple of a
and b is 42. The LCM of 5a and 11b is :
SSC CPO 09/11/2022 (Morning)
(a) 2310 (b) 4620 (c) 210 (d) 462
Q.97. The sum of two numbers is 1224
and their HCF is 68. The number of pairs
of numbers satisfying the above
condition is:
SSC CPO 09/11/2022 (Afternoon)
(a) 3 (b) 4 (c) 6 (d) 2
Q.98. The sides of a triangular ?eld are
62 m, 186 m and 279 m. Find the
greatest length of tape that would be
able to exactly measure each of them
without any fractions.
SSC CPO 10/11/2022 (Morning)
(a) 62 m (b) 93 m (c) 31 m (d) 30 m
Q.99. What is the LCM of
a
3
b - ab
3
, a
3
b
2
- a
2
b
3
, ab(a - b) ?
SSC CPO 11/11/2022 (Afternoon)
(a) a
2
b
2
(a
2
+ b
2
) (b) a
2
b
2
(a
2
- b
2
)
(c) a
2
b
3
(a
2
+ b
2
) (d) a
3
b
2
(a
2
- b
2
)
Q.100. The LCM and ratio of three
numbers are 1386 and 3 : 7 : 11,
respectively. The sum of the greatest and
least numbers is:
SSC CPO 11/11/2022 (Evening)
(a) 60 (b) 64 (c) 84 (d) 108
SSC MTS 2021 Tier - 1
Q.101. If and ?? = 2
8
× 3
5
, ?? = 2
3
× 3
4
,
, then what is the highest ?? = 3
5
× 2
7
common factor of P , Q and R?
SSC MTS 06/07/2022 (Afternoon)
(a) (b) 2
3
× 3
4
2
8
× 3
5
(c) (d) 2
2
× 3
2
2
4
× 3
5
Q.102. The tra?c lights at 3 different
road crossings change after every 48
sec, 72 sec and 108 sec, respectively. If
they all change simentionusly at 8 : 20
a.m., then at what will they next change
again simultaneously?
SSC MTS 07/07/2022 (Morning)
(a) 8 : 27 : 12 a.m. (b) 8 : 33 : 32 a.m.
(c) 8 : 12 : 18 a.m. (d) 8 : 40 : 14 a.m.
Q.103. The ratio of the LCM of two
numbers to the sum of the same two
numbers is 12 : 7. If their HCF is 4, what
is the product of these two numbers?
SSC MTS 14/07/2022 (Morning)
(a) 192 (b) 172 (c) 196 (d) 169
Q.104. P = 2
5
3
8
and Q = 2
3
3
K
. If the × ×
highest common factor of P and Q is 2
3
3
3
, then what is the value of K ? ×
SSC MTS 15/07/2022 (Morning)
(a) 1 (b) 2 (c) 3 (d) 5
Q.105. If and are two distinct prime ??
1
??
2
numbers, then what is the product of the
highest common factor and the least
common multiple of and ? ??
1
??
2
SSC MTS 20/07/2022 (Afternoon)
(a) 1 (b) (c) + (d)
??
1
??
2
??
1
??
2
??
1
× ??
2
Q.106. Two wires of lengths 10 m 54 cm
and 11 m 56 cm are both cut into pieces
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FAQs on SSC CGL Previous Year Questions (2023 - 18): HCF & LCM - SSC CGL Previous Year Papers
| 1. What is the difference between HCF and LCM? |  |
Ans.HCF (Highest Common Factor) is the largest number that divides two or more numbers without leaving a remainder, while LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers.
| 2. How can I calculate the HCF of two numbers using prime factorization? | |
Ans.To calculate the HCF using prime factorization, first, express both numbers as products of prime factors. Then, identify the common prime factors and multiply them together to get the HCF.
| 3. What is the relationship between HCF and LCM of two numbers? | |
Ans.The relationship between HCF and LCM of two numbers can be expressed as: HCF × LCM = Product of the two numbers. This means that if you know the HCF and one of the numbers, you can find the LCM easily.
| 4. How can I find the LCM of three or more numbers? | |
Ans.To find the LCM of three or more numbers, you can use the prime factorization method for each number and then take the highest power of each prime factor that appears in the factorizations. Multiply these together to get the LCM.
| 5. Are there any shortcuts to calculate HCF and LCM for large numbers? | |
Ans.Yes, for large numbers, you can use the Euclidean algorithm to find the HCF quickly. For LCM, you can use the formula LCM(a, b) = (a × b) / HCF(a, b) to simplify the calculation.
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Document Description: SSC CGL Previous Year Questions (2023 - 18): HCF & LCM for SSC CGL 2024 is part of SSC CGL Previous Year Papers preparation.
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