Solved Examples: HCF & LCM | Quantitative Aptitude for SSC CGL PDF Download

Definition of HCF

The concept of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) stands as one of the simplest and fundamental topics in mathematics to date. 

Definition of LCM

The notion of Highest Common Factor (HCF) and Lowest Common Multiple (LCM) remains one of the easiest and fundamental concepts in mathematics up to the present. 

HCF for Two Numbers

• When tasked with finding the HCF of two numbers, begin by dividing the larger number by the smaller one.
• Proceed to divide the divisor by the remainder.
• Continue this process of dividing the previous divisor by the remainder until a remainder of zero is achieved.
• The final divisor obtained through this process will be the HCF of the two given numbers.

HCF in case of more than two numbers

• The HCF of two numbers and the third number will be the HCF of all three numbers, extending to additional numbers in the set.
• Select any two numbers and determine their HCF using the method.

The rule for Solving HCF and LCM Questions and Answers

• Determine the multiples or factors of the larger number until identifying one that is also a multiple of the smaller number.
• Multiply all these identified factors, and the product will represent the LCM of the given numbers.

Examples

Example 1: Find the L.C.M of 15, 30, 45
(a) 90
(b) 95
(c) 92
(d) None of the above
Ans: (a)
2 | 15, 30, 45
3 | 15, 15, 45
3 | 5, 5, 15
5 | 5, 5, 5
| 1, 1, 1
L.C.M = 2*3*3*5 = 90

Example 2: Find the L.C.M of 25, 35, and 55
(a) 1900
(b) 1990
(c) 1925
(d) None of the above
Ans: (c)
5 | 25, 35, 55
5 | 5, 7, 11
7 | 1, 7, 11
11 | 1, 1, 11
| 1, 1, 1
L.C.M = 5*5*7*11 = 1925

Example 3: If 20 is the HCF of two particular numbers and the other two factors of their LCM are 10 and 12, find the larger number?
(a) 220
(b) 210
(c) 240
(d) None of the above
Ans: (c)
20* 10 = 200
20*12 = 240
So the larger number will be 240
HCF of two numbers is the number that is a common factor for both numbers given
Here 20 is the common factor.
Other than this common factor, we also will have the product of uncommon factors for the two numbers (10 and 12 here).
The first number = 20*10 = 200
and second number = 20 × 12 = 240
The greatest of two numbers is definitely 20 × 12 = 240

Example 4: The two specific numbers are in the ratio 6:7, if the HCF of the given numbers is 30, what will be the numbers?
(a) 160, 180
(b) 180, 210
(c) 200,160
(d) None of the above
Ans: (b)
Let the numbers be 6y and 7y
HCF = 30
The numbers will be 6*30 = 180
and 7*30 = 210

Example 5: The HCF of two numbers is 45, and their LCM is 90, if one specific number is 9, find the other number.
(a) 450
(b) 435
(c) 426
(d) None of the above
Ans: (a)
HCF * LCM = Products of Numbers
45 * 90 = 9 * x
Another number will be = (45*90)/9 = 450

Example 6: The given ratio of two numbers is 3:2. If the L.C.M of them is 30, then calculate their sum.
(a) 34
(b) 25
(c) 55
(d) None of the above
Ans: (b)
Given:
Ratio of the two numbers = 3: 2
LCM of two numbers = 30
To find: Sum of the two numbers
The formula used: Product of two numbers = LCM × HCF
Let the numbers be 3x and 2x
Product of two numbers = LCM × HCF
The HCF of two numbers will be x as the numbers are in ratio due to which it c an be conclude that their will be a HCF factor of x also.
The two numbers are 3x,2x
Product of two numbers = LCM × HCF
(3x )(2x) = 30 (x)
6x² = 30x
6x = 30
x = 30/6
x = 5
The first number is 3x
3x = 3(5)
= 15
The first number is 15
The second number is 2x
2x = 2(5)
=10
The second number is 10
Therefore the two numbers are 15,10
The sum of two numbers = 15 + 10
= 25

Example 7: Calculate the HCF of 22 and 33
(a) 11
(b) 12
(c) 15
(d) None of the above
Ans: (a)
22 = 2*11
33 = 3*11
So the HCF will be 11

Example 8: The HCF  of three specific numbers 6, 12, and 18 is 24 , find the LCM?
(a) 55
(b) 54
(c) 67
(d) None of the above
Ans: (b)
(6*12*18)/24
The next number will be = 54

Example 9: Determine the largest length of the tape, which can measure tape of 5 cm, 7cm, and 13 cm?
(a) 1
(b) 7
(c) 13
(d) None of the above
Ans: (a)
As all these numbers have no factors and are considered as prime numbers so their HCF will be 1.

Example 10: Street lamps start at the interval of 5, 10, 15, 20, and 25 seconds, calculate the number in which the street lamps began together in 40 minutes.
(a) 9 times
(b) 10 times
(c) 16 times
(d) None of the above
Ans: (c)

To solve this problem, we need to determine the least common multiple (LCM) of the intervals at

which the street lamps start: 5, 10, 15, 20, and 25 seconds. The LCM of these numbers will give us

the interval at which all the street lamps start together.

Let's find the LCM of 5, I0, 15, 20, and 25 seconds.

1. Prime factorization:

• 5=5 
• 10= 2 x 5
• 15=3 x 5
• 20 = 2² x 5
• 25= 5²

2. Take the highest power of each prime that appears in any factorization:

For 2: 2²

For 3: 3

For 5: 5²