Table of contents  
Exercise 3.1  
Exercise 3.2  
Exercise 3.3  
Exercise 3.4  
Exercise 3.5  
Exercise 3.6  
Exercise 3.7  
Factors and multiples
Q1. Write all the factors of the following numbers:
(a) 24
(b) 15
(c) 21
(d) 27
(e) 12
(f) 20
(g) 18
(h) 23
(i) 36
Ans:
(a) 24 = 1 × 24
24 = 2 × 12
24 = 3 × 8
24 = 4 × 6
24 = 6 × 4
∴ Factors of 24 = 1, 2, 3, 4, 6, 12, 24
(b) 15 = 1 × 15
15 = 3 × 5
15 = 5 × 3
∴ Factors of 15 = 1, 3, 5, 15
(c) 21 = 1 × 21
21 = 3 × 7
21 = 7 × 3
∴ Factors of 21 = 1, 3, 7, 21
(d) 27 = 1 × 27
27 = 3 × 9
27 = 9 × 3
∴ Factors of 27 = 1, 3, 9, 27
(e) 12 = 1 × 12
12 = 2 × 6
12 = 3 × 4
12 = 4 × 3
∴ Factors of 12 = 1, 2, 3, 4, 6, 12
(f) 20 = 1 × 20
20 = 2 × 10
20 = 4 × 5
20 = 5 × 4
∴ Factors of 20 = 1, 2, 4, 5, 10, 20
(g) 18 = 1 × 18
18 = 2 × 9
18 = 3 × 6
∴ Factors of 18 = 1, 2, 3, 6, 9, 18
(h) 23 = 1 × 23
∴ Factors of 23 = 1, 23
(i) 36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9
36 = 6 × 6
∴ Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Q2. Write first five multiplies of:
(a) 5
(b) 8
(c) 9
Ans:
(a) 5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 4 = 20
5 × 5 = 25
∴ The first five multiples of 5 are 5, 10, 15, 20, and 25.
(b) 8 × 1 = 8
8 × 2 = 16
8 × 3 = 24
8 × 4 = 32
8 × 5 = 40
∴ The first five multiples of 8 are 8, 16, 24, 32, and 40.
(c) 9 × 1 = 9,
9 × 2 = 18,
9 × 3 = ,27,
9 × 4 = 36,
9 × 5 = 45
∴ The first five multiples of 9 are 9, 18, 27, 36, and 45.
Q3. Match the items in column 1 with the items in column 2.
Ans:
(i) → (b)
The number 35 is a multiple of 7 because it can be divided evenly by 7.
(ii) → (d)
The number 15 is a factor of 30 because when we divide 30 by 15, we get a whole number without any remainder.
(iii) → (a)
The number 16 is a multiple of 8 because it can be divided evenly by 8.
(iv) → (f)
The number 20 is a factor of 20 because when we divide 20 by 20, we get a whole number without any remainder.
(v) → (e)
The number 25 is a factor of 50 because when we divide 50 by 25, we get a whole number without any remainder.
Q4. Find all the multiples of 9 up to 100.
Ans: 9 × 1 = 9
9 × 2 = 18
9 × 3 = 27
9 × 4 = 36
9 × 5 = 45
9 × 6 = 54
9 × 7 = 63
9 × 8 = 72
9 × 9 = 81
9 × 10 = 90
9 × 11 = 99
Multiples of 9 up to 100 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
Q2. State whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2, the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number.
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
Ans:
(a) False
Example: 3 + 5 + 7 = 15 (odd)
(b) True
Example: 3 + 5 + 6 = 14 (even)
(c) True
Example: 5 x 7 x 9 = 315 (odd)
(d) False
Example: 6 + 2 = 3 (odd)
(e) False
2 is a prime number but it is even
(f) False
3 is a prime number having 1 and 3 as its factors
(g) False
Example: 7 + 2 = 9 (odd)
(h) True
2 is even and the lowest prime number
(i) False
2 is even but not composite number
(j) True
Example: 4 x 6 = 24 (even)
Q3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.
Ans: 17 and 71; 37 and 73; 79 and 97
Q4. Write down separately the prime and composite numbers less than 20.
Ans: Prime numbers which are less than 20: 2, 3, 5, 7, 11, 13, 17, 19
Composite numbers which are less than 20: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
Q5. What is the greatest prime number between 1 and 10?
Ans: 2, 3, 5 and 7 are the prime numbers between 1 and 10. 7 is the greatest prime number among them.
Q6. Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18
Ans:
(a) 3 + 41 = 44
(b) 5 + 31 = 36
(c) 7 + 17 = 24
(d) 7 + 11 = 18
Q7. Give three pairs of prime numbers whose difference is 2.
[Remark: Two prime numbers whose difference is 2 are called twin primes].
Ans:
3 and 5
5 and 7
11 and 13
Q8. Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26
Ans: (a) 23 = 1 × 23 = 23 , 23 × 1 = 23
(b) 51 = 1 × 51 = 51 , 3 × 17 = 51, 17 × 3 = 51 , 51 × 1 = 51
1,3,17,51 they are not prime number.
(c) 37 = 1 × 37 = 37 , 37 × 1 = 37. They are prime number.
(d) 26 = 1 × 26 = 26, 2 × 13 = 26, 13 × 2 = 26, 26 × 1 = 26. They are not prime number.
Q9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Ans: 90 , 91 , 92 , 93 , 94 ,95 and 96 are seven consecutive composite number less than 100 and there is no prime number between them.
Q10. Express each of the following numbers as the sum of three odd primes:
(a) 21
(b) 31
(c) 53
(d) 61
Ans:
(a) 21 = 3 + 7 + 11
(b) 31 = 3 + 11 + 17
(c) 53 = 13 + 17 + 23
(d) 61 = 19 + 29 + 13
Q11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
(Hint: 3 + 7 = 10)
Ans:
(i) 2 + 3 =5
(ii) 2 + 13 = 15
(iii) 3 + 17 = 20
(iv) 7 +13 = 20
(v) 19 + 11 = 30
Q12: Fill in the blanks:
(a) A number which has only two factors is called a _______________.
(b) A number which has more than two factors is called a _______________.
(c) 1 is neither _______ nor _______.
(d) The smallest prime number is _______________.
(e) The smallest composite number is _______________.
(f) The smallest even number is _______________.
Ans:
(a) A number which has only two factors is called a Prime number.
A prime number is a number that can only be divided evenly by 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.
(b) A number which has more than two factors is called a Composite number.
A composite number is a number that has more than two factors. In other words, it can be divided evenly by more than just 1 and itself. For example, 4, 6, 8, and 9 are composite numbers.
(c) 1 is neither a prime number nor a composite number.
1 is a special number that is neither prime nor composite because it has only one factor, which is 1.
(d) The smallest prime number is 2.
The smallest prime number is 2 because it is the only even prime number. All other even numbers greater than 2 are divisible by 2 and therefore not prime.
(e) The smallest composite number is 4.
The smallest composite number is 4 because it can be divided evenly by 1, 2, and 4.
(f) The smallest even number is 2.
The smallest even number is 2 because it is the only even number that is not divisible by any other even number.
Number  Divisible by  
 2  3  4  5  6  8  9  10  11 
128  Yes  No  Yes  No  No  Yes  No  No  No 
990  
1586  
275  
6686  
639210  
429714  
2856  
3060  
406839 
Ans:
Number  Divisible by  
2  3  4  5  6  8  9  10  11  
128  Yes  No  Yes  No  No  Yes  No  No  No 
990  Yes  Yes  No  Yes  Yes  No  Yes  Yes  Yes 
1586  Yes  No  No  No  No  No  No  No  No 
275  No  No  No  Yes  No  No  No  No  Yes 
6686  Yes  No  No  No  No  No  No  No  No 
639210  Yes  Yes  No  Yes  Yes  No  No  Yes  Yes 
429714  Yes  Yes  No  No  Yes  No  Yes  No  No 
2856  Yes  Yes  Yes  No  Yes  Yes  No  No  No 
3060  Yes  Yes  Yes  Yes  Yes  No  Yes  Yes  No 
406839  No  Yes  No  No  No  no  No  No  No 
Q2. Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no):
Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150
Ans:
(a) 572
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(b) 726352
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(c) 5500
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(d) 6000
Divisible by 4 as its last two digits are 0.
Divisible by 8 as its last three digits are 0.
(e) 12159
Not divisible by 4 and 8 as it is an odd number.
(f) 14560
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(g) 21084
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(h) 31795072
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(i) 1700
Divisible by 4 as its last two digits are 0.
Not divisible by 8 as its last three digits are not divisible by 8.
(j) 5500
Not divisible by 4 as its last two digits are not divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
Q3. Using divisibility test, determine which of the following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852
Ans:
(a) 297144
Divisible by 2 as its units place is an even number.
Divisible by 3 as the sum of its digits (= 27) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is also divisible by 6.
(b) 1258
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 16) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(c) 4335
Not divisible by 2 as its units place is not an even number.
Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by 2, therefore, it is not divisible by 6.
(d) 61233
Not divisible by 2 as its units place is not an even number.
Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(e) 901352
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 20) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(f) 438750
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 27) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(g) 1790184
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 30) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(h) 12583
Not divisible by 2 as its units place is not an even number.
Not divisible by 3 as sum of its digits (= 19) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(i) 639210
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 21) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(j) 17852
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 23) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
Q4. Using divisibility test, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153
Ans:
(a) 5445
Sum of the digits at odd places = 4 + 5 = 9
Sum of the digits at even places = 4 + 5 = 9
Difference of both sums = 9 – 9 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(b) 10824
Sum of the digits at odd places = 4 + 8 +1 = 13
Sum of the digits at even places = 2 + 0 = 2
Difference of both sums = 13 – 2 = 11
Since the difference is 11, therefore, the number is divisible by 11.
(c) 7138965
Sum of the digits at odd places = 5 + 9 + 3 + 7 = 24
Sum of the digits at even places = 6 + 8 + 1 = 15
Difference of both sums = 24 – 15 = 9
Since the difference is neither 0 nor 11, therefore, the number is not divisible by 11.
(d) 70169308
Sum of the digits at odd places = 8 + 3 + 6 + 0 = 17
Sum of the digits at even places = 0 + 9 + 1 + 7 = 17
Difference of both sums = 17 – 17 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(e) 10000001
Sum of the digits at odd places = 1 + 0 + 0 + 0 = 1
Sum of the digits at even places = 0 + 0 + 0 + 1 = 1
Difference of both sums = 1 – 1 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(f) 901153
Sum of the digits at odd places = 3 + 1 + 0 = 4
Sum of the digits at even places = 5 + 1 + 9 = 15
Difference of both sums = 15 – 4 = 11
Since the difference is 11, therefore, the number is divisible by 11.
Q5. Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 3:
(a) __________ 6724
(b) 4765 __________ 2
Ans:
(a) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore, Smallest digit : 2 → 26724 = 2 + 6 + 7 + 2 + 4 = 21
Largest digit: 8 → 86724 = 8 + 6 + 7 + 2 + 4 = 27
(b) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore,
Smallest digit: 0 → 476502 = 4 + 7 + 6 + 5 + 0 + 2 = 24
Largest digit: 9 → 476592 = 4 + 7 + 6 + 5 + 0 + 2 = 33
Q6. Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 11:
(a) 92 __________ 389
(b) 8 __________ 9484
Ans:
(a) We know that a number is divisible by 11 if the difference of the sum of the digits at odd places and that of even places should be either 0 or 11.
Therefore, 928389 →
Odd places = 9 + 8 + 8 = 25
Even places = 2 + 3 + 9 = 14
Difference = 25 – 14 = 11
(b) We know that a number is divisible by 11 if the difference of the sum of the digits at odd places and that of even places should be either 0 or 11.
Therefore, 869484 →
Odd places = 8 + 9 + 8 = 25
Even places = 6 + 4 + 4 = 14
Difference = 25 – 14 = 11
Example for finding common factors
Q1. Find the common factors of:
(a) 20 and 28
(b) 15 and 25
(c) 35 and 50
(d) 56 and 120
Ans: (a) Factors of 20 = 1, 2, 4, 5, 10, 20
Factors of 28 = 1, 2, 4, 7, 14, 28
Common factors = 1, 2, 4
(b) Factors of 15 = 1, 3, 5, 15
Factors of 25 = 1, 5, 25
Common factors = 1, 5
(c) Factors of 35 = 1, 5, 7, 35
Factors of 50 = 1, 2, 5, 10, 25, 50
Common factors = 1, 5
(d) Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56
Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 60, 120
Common factors = 1, 2, 4, 8
Q2. Find the common factors of:
(a) 4, 8 and 12
(b) 5, 15 and 25
Ans: (a) Factors of 4 = 1, 2, 4
Factors of 8 = 1, 2, 4, 8
Factors of 12 = 1, 2, 3, 4, 6, 12
Common factors of 4, 8 and 12 = 1, 2, 4
(b) Factors of 5 = 1, 5
Factors of 15 = 1, 3, 5, 15
Factors of 25 = 1, 5, 25
Common factors of 5, 15 and 25 = 1, 5
Q3. Find the first three common multiples of:
(a) 6 and 8
(b) 12 and 18
Ans: (a) Multiple of 6 = 6, 12, 18, 24, 30, 36, 42, 28, 54, 60, 72, …………
Multiple of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, …………………….
Common multiples of 6 and 8 = 24, 48, 72
(b) Multiple of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ………
Multiple of 18 = 18, 36, 54, 72, 90, 108, ………………………………
Common multiples of 12 and 18 = 36, 72, 108
Q4. Write all the numbers less than 100 which are common multiples of 3 and 4.
Ans: Multiple of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99
Multiple of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
Common multiples of 3 and 4 = 12, 24, 36, 48, 60, 72, 84, 96
Q5. Which of the following numbers are coprime:
(a) 18 and 35
(b) 15 and 37
(c) 30 and 415
(d) 17 and 68
(e) 216 and 215
(f) 81 and 16
Ans: (a) Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 35 = 1, 5, 7, 35
Common factor = 1
Since, both have only one common factor, i.e., 1, therefore, they are coprime numbers.
(b) Factors of 15 = 1, 3, 5, 15
Factors of 37 = 1, 37
Common factor = 1
Since, both have only one common factor, i.e., 1, therefore, they are coprime numbers.
(c) Factors of 30 = 1, 2, 3, 5, 6, 15, 30
Factors of 415 = 1, 5, …….., 83, 415
Common factor = 1, 5
Since, both have more than one common factor, therefore, they are not coprime numbers.
(d) Factors of 17 = 1, 17
Factors of 68 = 1, 2, 4, 17, 34, 86
Common factor = 1, 17
Since, both have more than one common factor, therefore, they are not coprime numbers.
(e) Factors of 216 = 1, 2, 3, 4, 6, 8, 36, 72, 108, 216
Factors of 215 = 1, 5, 43, 215
Common factor = 1
Since, both have only one common factor, i.e., 1, therefore, they are coprime numbers.
(f) Factors of 81 = 1, 3, 9, 27, 81
Factors of 16 = 1, 2, 4, 8, 16
Common factor = 1
Since, both have only one common factor, i.e., 1, therefore, they are coprime numbers.
Q6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?
Ans: Factor of 5 = 1,5
Factor of 12 = 1, 2, 3, 4, 6, 12
Common factor = 1
And the product of these two number is 60.
Factor of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Q7. A number is divisible by 12. By what other numbers will that number be divisible?
Ans: Since the number is divisible by 12. Hence, it also divisible by its factors i.e., 1, 2, 3, 4, 6, 12
Therefore 1, 2, 3, 4, and 6 are the numbers other than 12 by which this number is also divisible
(b)
Ans:
(a) Since 6 = 2 × 3 and 10 = 5 × 2
(b) Since 60 = 30 × 2
30 = 10 × 3
10 = 5 × 2
Q2. Which factors are not included in the prime factorization of a composite number?
Ans: 1 is the factor which is not included in the prime factorization of a composite number.
Q3. Write the greatest 4digit number and express it in terms of its prime factors.
Ans: The greatest 4digit number = 9999The prime factors of 9999 are 3 × 3 × 11 × 101.
Q4. Write the smallest 5digit number and express it in terms of its prime factors.
Ans: The smallest five digit number is 10000.
The prime factors of 10000 are 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5.
Q5. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any, between, two consecutive prime numbers.
Ans: Prime factors of 1729 are 7 × 13 × 19.
The difference of two consecutive prime factors is 6.
Q6. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Ans: Among the three consecutive numbers, there must be one even number and one multiple of 3. Thus, the product must be multiple of 6.
Example: (i) 2 x 3 x 4 = 24
(ii) 4 x 5 x 6 = 120
Q7. The sum of two consecutive odd numbers is always divisible by 4. Verify this statement with the help of some examples.
Ans: 3 + 5 = 8 and 8 is divisible by 4.
5 + 7 = 12 and 12 is divisible by 4.
7 + 9 = 16 and 16 is divisible by 4.
9 + 11 = 20 and 20 is divisible by 4.
Q8. In which of the following expressions, prime factorization has been done:
(a) 24 = 2 x 3 x 4
(b) 56 = 7 x 2 x 2 x 2
(c) 70 = 2 x 5 x 7
(d) 54 = 2 x 3 x 9
Ans: (a) 24 = 2 × 3 × 4
Since 4 is composite. Hence, prime factorisation has not been done
(b) 56 = 7 × 2 × 2 × 2
Since all the factors are prime. Hence, prime factorisation has been done
(c) 70 = 2 × 5 × 7
Since all the factors are prime. Hence, prime factorisation has been done
(d) 54 = 2 × 3 × 9
Since 9 is composite. Hence prime factorisation has not been done
Q9. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.
Ans: No. Number 12 is divisible by both 6 and 4 but 12 is not divisible by 24.
Q10. I am the smallest number, having four different prime factors. Can you find me?
Ans: The smallest four prime numbers are 2, 3, 5 and 7.
Hence, the required number is 2 x 3 x 5 x 7 = 210
Example for calculating H.C.F.
Q1. Find the H.C.F. of the following numbers:
(a) 18, 48
(b) 30, 42
(c) 18, 60
(d) 27, 63
(e) 36, 84
(f) 34, 102
(g) 70, 105, 175
(h) 91, 112, 49
(i) 18, 54, 81
(j) 12, 45, 75
Ans: (a) Factors of 18 = 2 x 3 x 3
Factors of 48 = 2 x 2 x 2 x 2 x 3
H.C.F. (18, 48) = 2 x 3 = 6
(b) Factors of 30 = 2 x 3 x 5
Factors of 42 = 2 x 3 x 7
H.C.F. (30, 42) = 2 x 3 = 6
(c) Factors of 18 = 2 x 3 x 3
Factors of 60 = 2 x 2 x 3 x 5
H.C.F. (18, 60) = 2 x 3 = 6
(d) Factors of 27 = 3 x 3 x 3
Factors of 63 = 3 x 3 x 7
H.C.F. (27, 63) = 3 x 3 = 9
(e) Factors of 36 = 2 x 2 x 3 x 3
Factors of 84 = 2 x 2 x 3 x 7
H.C.F. (36, 84) = 2 x 2 x 3 = 12
(f) Factors of 34 = 2 x 17
Factors of 102 = 2 x 3 x 17
H.C.F. (34, 102) = 2 x 17 = 34
(g) Factors of 70 = 2 x 5 x 7
Factors of 105 = 3 x 5 x 7
Factors of 175 = 5 x 5 x 7
H.C.F. = 5 x 7 = 35
(h) Factors of 91 = 7 x 13
Factors of 112 = 2 x 2 x 2 x 2 x 7
Factors of 49 = 7 x 7
H.C.F. = 1 x 7 = 7
(i) Factors of 18 = 2 x 3 x 3
Factors of 54 = 2 x 3 x 3 x 3
Factors of 81 = 3 x 3 x 3 x 3
H.C.F. = 3 x 3 = 9
(j) Factors of 12 = 2 x 2 x 3
Factors of 45 = 3 x 3 x 5
Factors of 75 = 3 x 5 x 5
H.C.F. = 1 x 3 = 3
Q2. What is the H.C.F. of two consecutive:
(a) numbers?
(b) even numbers?
(c) odd numbers?
Ans: (a) The HCF of two consecutive numbers is 1
Example: The HCF of 2 and 3 is 1
(b) The HCF of two consecutive even numbers is 2
Example: The HCF of 2 and 4 is 2
(c) The HCF of two consecutive odd numbers is 1
Example: The HCF of 3 and 5 is 1
Q3. HCF of coprime numbers 4 and 15 was found as follows by factorisation : 4 = 2 × 2 and 15 = 3 × 5 since there is no common prime factor, so HCF of 4 and 15 is 0. Is the answer correct? If not, what is the correct HCF?
Ans: No. The correct H.C.F. is 1.
Q2. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?
Ans: Step measure by first boy = 63 cm
Step measure by second boy = 70 cm
Step measure by third boy = 77 cm
LCM = 2 × 3 × 3 × 5 × 7 × 11 = 6930
The minimum distance each should cover so that all can cover the distance in complete steps is 6930 cm.
Q3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
Ans: The measurement of longest tape = H.C.F. of 825 cm, 675 cm and 450 cm.
Factors of 825 = 3 x 5 x 5 x 11
Factors of 675 = 3 x 5 x 5 x 3 x 3
Factors of 450 = 2 x 3 x 3 x 5 x 5
H.C.F. = 3 x 5 x 5 = 75 cm
Therefore, the longest tape is 75 cm.
Q4. Determine the smallest 3digit number which is exactly divisible by 6, 8 and 12.
Ans:
LCM of 6,8,12 = 2 × 2 × 2 × 3 × = 24
Hence, the smallest 3digit number which meets the requirements must be a multiple of 24.
The smallest 3 – digit multiple of 24 is 24× 5 = 120.
Q5. Determine the greatest 3digit number exactly divisible by 8, 10 and 12.
Ans:
LCM = 2 × 2 × 2 × 3 × 5 = 120
Now we need to find the greatest 3digit multiple of 120
We may find 120 × 8 = 960 and 120 × 9 = 1080
Hence, 960 is the greatest 3digit number exactly divisible by 8, 10 and 12
Q6. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?
Ans:
LCM = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432
Hence, lights will change together after every 432 seconds
Therefore the lights will change simultaneously at 7 minutes and 12 seconds.
Q7. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.
Ans: The maximum capacity of container = H.C.F. (403, 434, 465)
Factors of 403 = 13 x 31
Factors of 434 = 2 x 7 x 31
Factors of 465 = 3 x 5 x 31
H.C.F. = 31
Therefore, 31 litres of the container is required to measure the quantity.
Q8. Find the least number which when divided by 6, 15 and 18, leave remainder 5 in each case.
Ans: LCM of 6, 15, 18
LCM = 2 × 3 × 3 × 5 = 90
Required number = 90 + 5
= 95
Hence, 95 is the required number.
Q9. Find the smallest 4digit number which is divisible by 18, 24 and 32.
Ans: LCM of 18, 24, 32
LCM = 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288
Here, we need to find the smallest 4digit multiple of 288
We find 288 × 3 = 864 and 288 × 4 = 1152
Hence, 1152 is the smallest 4digit number which is divisible by 18, 24 and 32
Q10. Find the L.C.M. of the following numbers:
(a) 9 and 4
(b) 12 and 5
(c) 6 and 5
(d) 15 and 4
Observe a common property in the obtained L.C.Ms. Is L.C.M. the product of two numbers in each case?
Ans: (a) L.C.M. of 9 and 4
= 2 x 2 x 3 x 3
= 36
(b) L.C.M. of 12 and 5
= 2 x 2 x 3 x 5
= 60
(c) L.C.M. of 6 and 5
= 2 x 3 x 5
= 30
(d) L.C.M. of 15 and 4
= 2 x 2 x 3 x 5
= 60
Yes in each case the LCM of given numbers is the product of these numbers.
Q11. Find the L.C.M. of the following numbers in which one number is the factor of other:
(a) 5, 20
(b) 6, 18
(c) 12, 48
(d) 9, 45
What do you observe in the result obtained?
Ans: (a) L.C.M. of 5 and 20 = 2 x 2 x 5 = 20
(b) L.C.M. of 6 and 18
= 2 x 3 x 3
= 18
(c) L.C.M. of 12 and 48
= 2 x 2 x 2 x 2 x 3
= 48
(d) L.C.M. of 9 and 45
= 3 x 3 x 5
= 45
∴ Hence, in each case the LCM of given numbers is the larger number. When a number is a factor of other number then their LCM will be the larger number.
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