Quantitative Aptitude · Lesson 2 of 18 · Updated ·
HCF and LCM — Meaning, Fast Methods and Word Problems
- 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
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.