The basics
- HCF (GCD): the largest number that divides all given numbers.
- LCM: the smallest number divisible by all given numbers.
- Link for two numbers: HCF × LCM = product of the numbers.
- Method: prime-factorise. HCF takes the lowest power of each common prime; LCM takes the highest power of each prime.
Worked examples
Example 1. 36 = 2² × 3²; 48 = 2⁴ × 3. HCF = 2² × 3 = 12; LCM = 2⁴ × 3² = 144. Check: 12 × 144 = 1,728 = 36 × 48 ✓.
Example 2 (bells). Three bells ring at intervals of 12, 18 and 30 minutes. They next ring together after the LCM: 12 = 2² × 3; 18 = 2 × 3²; 30 = 2 × 3 × 5; LCM = 2² × 3² × 5 = 180 minutes.
Example 3 (largest tile). A room 420 cm by 300 cm is to be paved with square tiles of the largest possible size. The side is the HCF of 420 and 300 = 60 cm. Number of tiles: (420 ÷ 60) × (300 ÷ 60) = 7 × 5 = 35.
Example 4 (remainders). The greatest number that divides 70 and 105 leaving remainders 4 and 6? Subtract the remainders: 66 and 99; HCF = 33. Check: 70 = 33 × 2 + 4 ✓; 105 = 33 × 3 + 6 ✓.
Example 5 (smallest number with the same remainder). The smallest number which, when divided by 6, 8 and 12, leaves remainder 5 each time: LCM(6, 8, 12) = 24; the number is 24 + 5 = 29.
Example 6 (fractions). HCF of 2/3, 4/9 and 6/15 = HCF of numerators ÷ LCM of denominators = 2 ÷ 45 (taking 6/15 as 2/5, numerators 2, 4, 2; denominators 3, 9, 5; LCM 45). So 2/45.
Common traps
- Mixing up HCF and LCM in word problems: "together again" means LCM; "largest equal groups or pieces" means HCF.
- Forgetting to subtract remainders in "leaves remainder" problems when finding HCF, or to add them when finding LCM.
- Using the product identity for three numbers. It holds only for two.
- Not reducing fractions before taking HCF or LCM.
Quick tests
- If one number divides another, the HCF is the smaller and the LCM the larger.
- For co-prime numbers, the HCF is 1 and the LCM is their product.
- Divisibility rules (2, 3, 4, 5, 9, 11) speed factorisation.
A practice routine
- Day 1: prime factorisation and direct HCF and LCM, ten questions.
- Day 2: word problems (bells, tiles, groups).
- Day 3: remainder problems and fractions.
- Day 4: a timed set of 12 at 50 seconds each.
Where it appears
HCF and LCM appear in SSC CGL, railway mathematics, banking and MAT papers, and support time and work problems.
Practice tests below are published by institutes and educators on TestMint.
