# HCF and LCM — Meaning, Fast Methods and Word Problems

Source: https://skills101.in/learn/aptitude/hcf-and-lcm — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

- **HCF** (Highest Common Factor, also called GCD) = the **biggest number that divides** all the given numbers.
- **LCM** (Least Common Multiple) = the **smallest number that all the given numbers divide**.

Example with 12 and 18:

- Factors of 12: 1, 2, 3, 4, **6**, 12. Factors of 18: 1, 2, 3, **6**, 9, 18 → HCF = **6**
- Multiples of 12: 12, 24, **36**, … Multiples of 18: 18, **36**, … → LCM = **36**

## The prime-factor method (fastest)

Break each number into primes. Then:

- **HCF** = the primes they **share**, each with the **smaller** power
- **LCM** = **every** prime that appears, each with the **bigger** power

![Venn diagram: 12 = 2×2×3 and 18 = 2×3×3 share 2 and 3, so HCF = 6 and LCM = 2×2×3×3 = 36](https://skills101.in/learn-img/aptitude/hcf-lcm-venn.svg)

### Example 1
*Find the HCF and LCM of 72 and 120.*

- 72 = 2³ × 3²
- 120 = 2³ × 3 × 5
- HCF = 2³ × 3 = **24** (shared primes, smaller powers)
- LCM = 2³ × 3² × 5 = **360** (all primes, bigger powers)

## The golden rule for two numbers

> **HCF × LCM = product of the two numbers**

Check: 24 × 360 = 8640 and 72 × 120 = 8640 ✓

### Example 2
*The HCF of two numbers is 12 and their LCM is 360. One number is 60. Find the other.*

Other = (12 × 360) ÷ 60 = **72**.

(This rule 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: LCM of 2/3 and 4/9 = LCM(2, 4) ÷ HCF(3, 9) = 4 ÷ 3 = **4/3**.

## Word problems: HCF or LCM?

| Clue in the question | Use |
|---|---|
| "**largest** size / length / number that divides…", cutting into equal pieces, filling with the biggest tile | **HCF** |
| "**smallest** number divisible by…", bells ringing **together again**, people meeting again | **LCM** |

### Example 3 — bells
*Three bells ring every 6, 8 and 12 minutes. They ring together at 9:00 am. When do they next ring together?*

They meet again after LCM(6, 8, 12) = **24 minutes** → **9:24 am**.

### Example 4 — cutting
*Two ropes are 84 m and 126 m long. They must be cut into pieces of equal length, as long as possible. How long is each piece, and how many pieces are there?*

Length = HCF(84, 126) = **42 m**. Pieces = 84/42 + 126/42 = 2 + 3 = **5**.

### Example 5 — same remainder
*Find the smallest number that leaves remainder 3 when divided by 4, 6 and 9.*

LCM(4, 6, 9) = 36. Add the remainder: 36 + 3 = **39**. (Check: 39 = 4×9+3 = 6×6+3 = 9×4+3 ✓)

### Example 6 — largest divisor with the same remainder
*Find the largest number that divides 43, 91 and 183 leaving the same remainder each time.*

Take differences: 91 − 43 = 48, 183 − 91 = 92, 183 − 43 = 140. HCF(48, 92, 140) = **4**.

## Common mistakes

- Using HCF × LCM = product for **three** numbers. It only works for two.
- Mixing up the clue words: "largest" usually means HCF, "smallest… divisible by" means LCM.
- In "same remainder" questions, forgetting to take **differences** first.

## Quick recap

- HCF: shared primes, smaller powers. LCM: all primes, bigger powers.
- For two numbers: HCF × LCM = product.
- "Equal pieces / biggest" → HCF. "Together again / smallest divisible" → LCM.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/hcf-and-lcm