HCF and LCM: Concepts, Formulas, and Clear Examples

For positive integers, HCF is the largest common divisor; LCM is the smallest common multiple. Product links them: $\text{HCF}\times\text{LCM}=a\times b$ for two numbers.

Prime factorization is the universal tool; division method finds HCF fast for large pairs.

Word problems hide HCF as largest equal grouping and LCM as first common time/event.

Prime factorization

HCF

$$\text{HCF} = \prod p_i^{\min(\alpha_i,\beta_i)}$$

LCM

$$\text{LCM} = \prod p_i^{\max(\alpha_i,\beta_i)}$$

Product identity (two numbers)

Key link

$$\text{HCF}(a,b)\times\text{LCM}(a,b) = a\times b$$

Example — Find LCM from HCF

HCF(12,18)=6, product 216. LCM = $216/6=36$.

Euclidean algorithm

• HCF(a,b) = HCF(b, a mod b) until remainder 0.
• Last non-zero divisor is HCF.

Co-prime check

• HCF = 1 ⇒ numbers co-prime.
• LCM of co-prime pair = product.

Three or more numbers

• HCF/LCM extend by taking min/max exponents across all factorizations.
• LCM of fractions: $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$.

Application patterns

• Largest tile size covering floor → HCF of dimensions.
• Bells ring together → LCM of periods.

Step-by-step method for exam questions

1. Factor each number (or use Euclidean for HCF).
2. HCF: min powers; LCM: max powers on each prime.
3. Verify product rule for two numbers.
4. Translate word problem to HCF or LCM.
5. Sanity-check HCF divides both numbers.
6. Select option.

Formula cheat sheet

Frequently asked questions

HCF of 0 and n?

Not in standard MCQs; assume positive integers.

LCM always ≥ larger number?

Yes for positive integers.

Fractions LCM?

Use $\text{LCM}(\frac{a}{b},\frac{c}{d})=\frac{\text{LCM}(a,c)}{\text{HCF}(b,d)}$.

Three numbers product rule?

No simple $abc$ product; factor all three.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.