Practice HCF and LCM Questions and Answers

The Highest Common Factor (HCF) and the Least Common Multiple (LCM) are a part of the Quantitative Aptitude syllabus for government competitive examinations for SBI PO, SBI Clerk, IBPS PO, IBPS Clerk, LIC AAO, SSC CGL, SSC CHSL and railways. The HCF is the greatest factor present between the given two or more numbers. It is the highest number that can be divided into 2 or more numbers without leaving any remainders. Whereas, LCM is the smallest factor by which two or more given numbers may exactly be divided. HCF is also referred to as the greatest common factor and LCM is also known as the least common divisor.

There are mainly 3 methods to derive the HCF and LCM of a given set of numbers. They are;

• Factorization method: 

HCF: Firstly, the multiple factors of all the given numbers are listed. Then, the highest factor which is common to all the numbers is the HCF.

E.g., find the HCF of 36 and 45. 

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

45: 1, 3, 5, 9, 45

While 1, 3 and 9 are all common to 36 and 45, the number of greatest value common to both is 9. 

HCF= 9. 

LCM: 

Find the LCM Of 25 and 30

25= 25, 50, 75, 100, 125, 150, 175, 200

30= 30, 60, 90, 120, 150, 180, 210
The lowest factor common to both is 150. 

∴ LCM= 150

• Long Division method: In a given set of numbers, consider the smallest number to the divisor and the larger number(s) as the dividend. Keep dividing using the long division method until the remainder achieved is 0. 

HCF: Find the HCF of 24 and 48

1

24⟌36 

2

 6 ⟌12

-12

0

∴ HCF= 6

LCM: 

Find the LCM of 24 and 36.

2⟌24, 36

2⟌12, 18

2⟌6, 9

2⟌3,9

3⟌1,3

3⟌1,1

LCM= 2x2x2x2x3x3= 144

∴LCM=144

• Prime Factorization method: List all the factors via which the provided numbers can be accurately divided to express all the numbers as a product of their prime numbers. The largest factor common to all the numbers is the HCF.

HCF: The Prime factor

E.g., find the HCF of 30, 36 and 42. 

30=2x3x5

36=2x2x3x3x3

45= 2x3x7

3 is the greatest number that divides all the given numbers. Thus 3 is the HCF of 30, 36 and 45.

LCM:

Find the LCM of 25 and 30

25= 5x5

30= 2x3x5

LCM= 5X2X3X5= 150

∴ LCM= 150

Relationship between HCF and LCM.

• The product of the HCF and LCM of any two given numbers is always equal to the product of the two given numbers. However, this rule applies to a pair of two numbers only and not more. Formula: product of 2 numbers = (HCF of the two numbers) x (LCM of the two numbers)
• Since the HCF of co-prime numbers is 1, the LCM of the given co-prime numbers is equal to the product of the numbers.
  Formula: LCM of co-prime numbers= product of the co-prime numbers. 

Of fractions;
HCF of fractions= HCF of numerator / LCM of denominator

LCM of fractions= LCM of Numerator / HCF of Denominator

While HCF and LCM questions are always assessed in multiple-choice question (MCQ) format, the pattern of questions asked may not always be the same. Rather than simply providing numbers and asking for their HCFs and LCMs, sometimes, the HCF and/or LCM may be provided with one or more numbers and the question might ask to find the other number from the set. 

E.g., The HCF of two numbers is 4 and the LCM of the same is 6528. If one number is 204, find the other number. 

• 240
• 128
• 328
• 186

We will use the prime factorization method to solve this question.

204= 2x2x3x17

The prime factors of the provided options are only 240

240= 2 × 2 × 2 × 2 × 3 × 5

128= 2x2x2x2x2x2x2

328= 2x2x2x41

186= 2x2x2x23

Thus, we can see that from the given options, only the 128 can have an HCF of 4 with 204. Therefore, the correct answer is option b. 

Applications of HCF and LCM.

HCF:

• Two divide something into groups and rows/columns.
• To split something into smaller portions.
• To equally distribute something into the largest group possible.

LCM:

• To buy/ purchase multiple items to have enough to be divided among a certain number.
• To analyze the repetitive pattern of events.