LCM and HCF: Methods, Tricks & Word Problems

LCM and HCF are two of the most useful ideas in school maths. The HCF (Highest Common Factor) is the biggest number that divides two or more numbers exactly. The LCM (Lowest Common Multiple) is the smallest number that all of them divide into exactly. HCF is also called GCD (Greatest Common Divisor).

Factors vs Multiples: A 10-Second Refresher

Factors of 12: 1, 2, 3, 4, 6, 12 (they divide 12 exactly). Multiples of 12: 12, 24, 36, 48… (12 divides them exactly). HCF works with factors; LCM works with multiples.

Method 1: Prime Factorisation

Write each number as a product of primes. Find the HCF and LCM of 36 and 48.

36 = 2² × 3²    48 = 2⁴ × 3

HCF = the lowest power of each common prime = 2² × 3 = 12
LCM = the highest power of every prime = 2⁴ × 3² = 16 × 9 = 144

Check: 36 × 48 = 1728, and HCF × LCM = 12 × 144 = 1728 ✓

Method 2: The Division Method (Best for LCM of 3 or More Numbers)

Find the LCM of 12, 15 and 20.

Divide by	12	15	20
2	6	15	10
2	3	15	5
3	1	5	5
5	1	1	1

Keep dividing by primes that divide at least one number, copying down any number that isn't divisible. LCM = 2 × 2 × 3 × 5 = 60

Method 3: Euclid's Division Algorithm (Class 10, Best for Large Numbers)

Divide the bigger number by the smaller one, then divide the divisor by the remainder. Repeat until the remainder is 0 — the last divisor is the HCF.

HCF of 252 and 105:
252 = 105 × 2 + 42
105 = 42 × 2 + 21
42 = 21 × 2 + 0
HCF = 21

Quick Tricks Worth Remembering

Trick	Example
For two numbers: HCF × LCM = product of the numbers	HCF 6, product 216 → LCM = 36
If one number divides the other, HCF = smaller and LCM = bigger	8 and 24 → HCF 8, LCM 24
Co-prime numbers have HCF 1 and LCM = product	8 and 15 → LCM 120
The HCF always divides the LCM	12 divides 144
HCF ≤ smallest number; LCM ≥ largest number	Use this to check your answer

Note: HCF × LCM = product works for two numbers only, not three.

HCF and LCM of fractions: HCF of fractions = HCF of numerators ÷ LCM of denominators; LCM of fractions = LCM of numerators ÷ HCF of denominators. Example: for 2/3 and 4/9, HCF = 2/9 and LCM = 4/3.

Word Problems: How to Know Which One to Use

Rule of thumb: "largest", "maximum", "greatest size", "equal pieces" → usually HCF. "Smallest", "least", "together again", "at the same time" → usually LCM.

1. Bells (LCM): Three bells ring every 6, 8 and 12 minutes, together at 9:00 am. LCM of 6, 8 and 12 = 24 minutes, so they next ring together at 9:24 am.

2. Tiles (HCF): A floor is 360 cm by 240 cm. The largest square tile that fits exactly = HCF of 360 and 240 = 120 cm.

3. Ribbons (HCF): Ribbons of 84 cm and 126 cm cut into equal pieces of the greatest length = HCF of 84 and 126 = 42 cm.

4. Remainders (LCM + remainder): Smallest number leaving remainder 5 when divided by 12, 15 or 18 = LCM(12,15,18) + 5 = 180 + 5 = 185.

5. Runners (LCM): Two runners take 4 and 6 minutes per round, starting together. LCM of 4 and 6 = 12 minutes until they meet again at the start.

Practice Questions

1. Find the HCF and LCM of 18 and 30.
2. Find the LCM of 8, 12 and 20.
3. Use Euclid's algorithm to find the HCF of 135 and 225.
4. The HCF of two numbers is 4 and their product is 480. Find their LCM.

Answers: 1. HCF 6, LCM 90   2. 120   3. 45   4. 120

Frequently Asked Questions

What is the difference between LCM and HCF?
HCF is the largest number that divides all the given numbers. LCM is the smallest number that all the given numbers divide into.

What is the relation between HCF and LCM?
For two numbers, HCF × LCM = the product of the two numbers.

Which method is best for finding HCF?
Prime factorisation is best for small numbers. Euclid's division algorithm is fastest for large numbers.

How do I know whether a word problem needs LCM or HCF?
If you are cutting or dividing things into the largest equal parts, use HCF. If you need the time or quantity when things happen together again, use LCM.