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LCM and HCF Questions with Solutions

LCM and HCF questions reward knowing the prime-factorization method cold, plus one shortcut connecting the two — once you have that, most questions in this topic become routine. Here’s the concept, four fully worked examples, and a few to try yourself.

The concept, quickly

  • HCF (Highest Common Factor) is the largest number that divides every given number exactly. LCM (Lowest Common Multiple) is the smallest number that every given number divides into exactly.
  • Prime factorization method: break each number into prime factors. HCF = product of the common prime factors, each at its LOWEST power across the numbers. LCM = product of all prime factors (common and uncommon), each at its HIGHEST power.
  • Useful shortcut for two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. This lets you find one directly from the other without refactorizing.
  • For "largest number that divides X, Y and Z leaving the same remainder R" questions, find the HCF of (X − R), (Y − R) and (Z − R) instead of X, Y, Z directly.

Worked examples

1. Find the HCF and LCM of 12 and 18.

Solution

  1. 12 = 2² × 3. 18 = 2 × 3².
  2. HCF = common factors at their lowest power = 2¹ × 3¹ = 6.
  3. LCM = all factors at their highest power = 2² × 3² = 4 × 9 = 36.
  4. Check: HCF × LCM = 6 × 36 = 216, and 12 × 18 = 216 — matches.

Answer: HCF = 6, LCM = 36

2. Find the HCF and LCM of 24, 36, and 60.

Solution

  1. 24 = 2³ × 3. 36 = 2² × 3². 60 = 2² × 3 × 5.
  2. HCF = common factors (2 and 3 appear in all three; 5 doesn’t) at their lowest power across all three: 2² × 3¹ = 4 × 3 = 12.
  3. LCM = all factors at their highest power: 2³ × 3² × 5¹ = 8 × 9 × 5 = 360.

Answer: HCF = 12, LCM = 360

3. Two numbers are in the ratio 3 : 4 and their LCM is 48. Find the HCF and the two numbers.

Solution

  1. Let the numbers be 3x and 4x. Since 3 and 4 share no common factor, x itself is the HCF, and the LCM is 3 × 4 × x = 12x.
  2. 12x = 48 → x = 4.
  3. The numbers are 3 × 4 = 12 and 4 × 4 = 16. HCF = x = 4. Check: HCF(12, 16) = 4 and LCM(12, 16) = 48 — matches.

Answer: HCF = 4, numbers are 12 and 16

4. Find the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively.

Solution

  1. The required number is the HCF of (70 − 5) and (125 − 8), i.e., HCF(65, 117).
  2. 65 = 5 × 13. 117 = 3² × 13. HCF = 13.
  3. Check: 70 ÷ 13 = 5 remainder 5. 125 ÷ 13 = 9 remainder 8 — both match.

Answer: 13

Try these yourself

Work through these the same way, then check against the answer.

1. Find the HCF and LCM of 15 and 20.

Answer: HCF = 5, LCM = 60

2. Find the HCF and LCM of 8, 12, and 20.

Answer: HCF = 4, LCM = 120

3. The HCF of two numbers is 6 and their LCM is 90. If one of the numbers is 18, find the other.

Answer: 30

Where this comes up

LCM and HCF is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk.

Practice more LCM and HCF questions

This page covers the concept and a handful of worked examples. Pariksha Saathi has full topic-wise practice sets for LCM and HCF (and every other Quant/Reasoning/English topic) with instant scoring and explanations — free, no account required.

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