How to Find LCM and HCF (Easy Methods)
LCM and HCF are two of the most useful ideas in number theory, and they appear in CBSE and ICSE exams from primary classes right up to Class 10. The good news is that finding LCM and HCF is easy once you know two reliable methods: prime factorisation and the division method. This guide explains both, with clear worked examples and a handy shortcut formula.
What are LCM and HCF?
- HCF (Highest Common Factor): the largest number that divides two or more numbers exactly. It is also called the GCD (Greatest Common Divisor).
- LCM (Lowest Common Multiple): the smallest number that is a multiple of two or more numbers.
For example, the HCF of 12 and 18 is 6, because 6 is the biggest number that divides both. The LCM of 12 and 18 is 36, because 36 is the smallest number both divide into.
Method 1: Prime factorisation
Break each number into its prime factors, then combine. Let us find the HCF and LCM of 12 and 18.
- Step 1: 12 = 2 × 2 × 3 = 2² × 3.
- Step 2: 18 = 2 × 3 × 3 = 2 × 3².
- Step 3 (HCF): multiply the common factors with the lowest powers: 2 × 3 = 6.
- Step 4 (LCM): multiply each prime with the highest power: 2² × 3² = 4 × 9 = 36.
So HCF = 6 and LCM = 36, matching what we saw above.
Method 2: Division method
For HCF, the long-division (Euclidean) method is quick for large numbers. To find the HCF of 48 and 60:
- Step 1: Divide 60 by 48; remainder is 12.
- Step 2: Divide 48 by 12; remainder is 0.
- Step 3: The last divisor giving zero remainder is 12, so HCF = 12.
For LCM, use short division by common primes. Write the numbers side by side and divide by primes until you reach 1, multiplying all divisors. For 12 and 18: divide by 2 to get 6 and 9, divide by 3 to get 2 and 3, divide by 2 and 3 separately to finish. The divisors 2 × 3 × 2 × 3 = 36, which is the LCM.
The LCM and HCF relationship
There is a beautiful shortcut for two numbers: LCM × HCF = product of the two numbers. Check it with 12 and 18: LCM × HCF = 36 × 6 = 216, and 12 × 18 = 216. They match!
This formula is a real time-saver. If you know any three of the four values, you can find the fourth. For instance, if two numbers are 8 and 12 with HCF 4, then LCM = (8 × 12) ÷ 4 = 96 ÷ 4 = 24. Note this rule applies to exactly two numbers, not three or more.
These skills support fractions work and ratios in Class 9 Maths and the real numbers chapter in Class 10 Maths. For quick mental shortcuts, see our fun maths tricks for students.
Frequently asked questions
What is the difference between LCM and HCF?
HCF is the largest number that divides the given numbers exactly, while LCM is the smallest number that all the given numbers divide into. HCF is always less than or equal to the smallest number; LCM is always greater than or equal to the largest.
Does the rule LCM × HCF = product work for three numbers?
No. The relationship LCM × HCF equals the product holds only for two numbers. For three or more numbers it does not generally apply, so use prime factorisation instead.
When do we use LCM and when do we use HCF?
Use LCM for problems about events repeating together or adding fractions with different denominators. Use HCF for splitting things into the largest equal groups or simplifying fractions to lowest terms.