HCF & LCM

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Subtopic 2 of 4·← HCF & LCM: definitions and core relations·Standard word problems (tiles, bells, groups, divisible numbers) →

Finding HCF & LCM (incl. fractions and decimals)

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Prime factorisation: HCF = product of common primes with the lowest powers; LCM = product of all primes with the highest powers.

Long division method is faster for two numbers: divide, then divide the divisor by the remainder, repeat; the last non-zero remainder is the HCF. LCM $= \frac{ab}{\text{HCF}}$ .

Fractions: $\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$ , $\quad\text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$ .

Decimals: equalise decimal places first (write 0.5 as 0.500), find HCF/LCM of the integers, put the decimal point back with the same number of places.

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 12 practice Q

Detailed notes

The two standard methods

1. Prime factorisation (best for two to four numbers). Write every number as a product of primes.

HCF = product of the primes common to all, each with its lowest power.
LCM = product of all primes that appear, each with its highest power. Example: $32 = 2^5$ , $36 = 2^2 \cdot 3^2$ , $45 = 3^2 \cdot 5$ → nothing is common to all three, so HCF $= 1$ ; LCM $= 2^5 \cdot 3^2 \cdot 5 = 1440$ . A small table helps: one row per number, one column per prime. HCF = read column minima; LCM = read column maxima.

2. Long (successive) division (fastest for two big numbers). Divide the bigger number by the smaller; then keep dividing the last divisor by the last remainder. The last non-zero divisor is the HCF. Then LCM $= \frac{a \times b}{\text{HCF}}$ . Example: HCF of 4052 and 12576: $12576 = 3 \times 4052 + 420$ → $4052 = 9 \times 420 + 272$ → $420 = 272 + 148$ → $272 = 148 + 124$ → $148 = 124 + 24$ → $124 = 5 \times 24 + 4$ → $24 = 6 \times 4 + 0$ . HCF $= 4$ . For three numbers: take the HCF of the first two, then the HCF of that value with the third.

Fractions: the cross rule

$\text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$ Example: HCF of $\frac{2}{3}, \frac{8}{9}, \frac{10}{27}$ is $\frac{\gcd(2, 8, 10)}{\text{LCM}(3, 9, 27)} = \frac{2}{27}$ . Why it works: writing all fractions over the LCM of the denominators turns them into whole numbers; the HCF (or LCM) of those whole numbers, divided back by that LCM of denominators, gives exactly this rule.

Decimals: shift the point, then shift it back

Multiply every number by the same power of 10 so all become integers (use as many zeros as the largest number of decimal places).
Find the HCF or LCM of the integers.
Put the decimal point back with that many places. Example: HCF of $0.54, 1.8, 7.2$ → work with $54, 180, 720$ → HCF $= 18$ → answer $0.18$ .

Numbers already given as prime products

If the numbers come as $2^3 \cdot 3 \cdot 5$ -style products, skip factorising: HCF takes the minimum power of each prime across the numbers, LCM the maximum. Example: HCF of $2^3 \cdot 3^2 \cdot 5^4$ and $2^2 \cdot 3^3 \cdot 5^2$ is $2^2 \cdot 3^2 \cdot 5^2 = 180$ .

Choosing a method fast

Small numbers (say up to 100): factorise mentally.
Two big numbers: long division.
Prime-product form: read min/max powers directly.
Fractions: cross rule. Decimals: clear the point, restore it at the end.

Quick revision

HCF: common primes, lowest powers. LCM: every prime, highest powers.
Division method: last non-zero divisor is the HCF; LCM = product ÷ HCF.
Fractions: HCF of tops over LCM of bottoms (and the mirror image for LCM).
Decimals: clear the decimal point first, restore the same number of places at the end.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: HCF by successive (long) divisioncommon3 practice Q

How to spot it:

Two or three larger numbers and the HCF is asked — the numbers are too big to factorise comfortably.

\gcd(a, b): \text{ divide } a \text{ by } b, \text{ then } b \text{ by the remainder, } \dots \text{ last non-zero divisor is the HCF}; \quad \text{LCM} = \frac{ab}{\gcd(a,b)}

Divide the bigger number by the smaller; note the remainder.
Divide the old divisor by the remainder; repeat until the remainder is 0.
The last non-zero divisor is the HCF. For a third number, repeat with (HCF, third number).
LCM of two numbers = product ÷ HCF.

Why: any common divisor of a and b also divides every remainder, so the process preserves the common divisors while shrinking the numbers.

Example: Find the HCF of 4052 and 12576.

$12576 = 3 \times 4052 + 420$ → $4052 = 9 \times 420 + 272$ → … → $24 = 6 \times 4 + 0$ . HCF $= 4$ .

Type 2: HCF / LCM from prime-power form (min vs max powers)common3 practice Q

How to spot it:

The numbers are given as prime products, like $2^3 \times 3^2 \times 5^4$ , or the HCF and LCM of power-form numbers are asked.

\text{HCF} = \prod p_i^{\min}, \qquad \text{LCM} = \prod p_i^{\max}

List each prime that appears anywhere.
HCF: for each prime take the SMALLEST exponent across all the numbers (skip primes missing from any number).
LCM: for each prime take the LARGEST exponent anywhere.
Multiply out only at the end.

Why: the HCF must fit inside every number; the LCM must contain every number.

Example: Find the HCF of $2^3 \times 3^2 \times 5^4$ , $2^2 \times 3^3 \times 5^2$ and $2^4 \times 3 \times 5^3$ .

Lowest powers: $2^2$ , $3^1$ , $5^2$ → HCF $= 4 \times 3 \times 25 = 300$ . (The LCM would be $2^4 \times 3^3 \times 5^4$ . )

Type 3: HCF and LCM of fractions (cross rule)very common2 practice Q

How to spot it:

A list of fractions like 2/3, 8/9, 10/27 with HCF or LCM asked.

\text{HCF} = \frac{\gcd(\text{numerators})}{\text{LCM}(\text{denominators})}, \qquad \text{LCM} = \frac{\text{LCM}(\text{numerators})}{\gcd(\text{denominators})}

Reduce each fraction to lowest terms first.
For the HCF: HCF of the tops, LCM of the bottoms.
For the LCM: LCM of the tops, HCF of the bottoms.
Sanity check: every given fraction ÷ HCF should be a whole number; LCM ÷ every fraction should be a whole number.

Why: putting all fractions over the LCM of denominators turns them into integers, and the ordinary HCF/LCM of those integers gives the rule.

Example: Find the HCF of $\frac{2}{3}$ , $\frac{8}{9}$ and $\frac{10}{27}$ .

$\frac{\gcd(2, 8, 10)}{\text{LCM}(3, 9, 27)} = \frac{2}{27}$ . Check: each fraction ÷ $\frac{2}{27}$ is whole (9, 12, 5).

Type 4: HCF and LCM of decimalscommon2 practice Q

How to spot it:

Decimals like 0.54, 1.8, 7.2 with HCF or LCM asked — answers have a decimal point.

\text{scale by } 10^k \to \gcd/\text{lcm of integers} \to \text{scale back by } 10^k

Count the largest number of decimal places among the values.
Multiply every value by that power of 10 so all become integers.
Find the HCF or LCM of the integers.
Divide the result by the same power of 10.

Why: multiplying everything by a common scale factor multiplies HCF and LCM by the same factor — so undo it at the end.

Example: Find the LCM of 0.12, 0.18 and 0.30.

× 100 → 12, 18, 30 → LCM 180 → restore two places → 1.80. Check: $1.8 \div 0.12 = 15$ ✓.

Formulas

HCF by factors

\text{HCF} = p_1^{\min} \cdot p_2^{\min} \cdots \text{(common primes, lowest powers)}

LCM by factors

\text{LCM} = p_1^{\max} \cdot p_2^{\max} \cdots \text{(all primes, highest powers)}

HCF of fractions

\text{HCF}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\gcd(a, c)}{\text{LCM}(b, d)}

LCM of fractions

\text{LCM}\!\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\gcd(b, d)}

Two-number division method

\text{LCM} = \frac{a \times b}{\gcd(a,b)}

Shortcut tricks

⚡ LCM row of prime powers

Write each number as a row of prime powers; LCM = read column maxima, HCF = read common minima. Three numbers take under 30 seconds.

Example: Find the LCM of 32, 36, 45.

$32 = 2^5$ , $36 = 2^2 \cdot 3^2$ , $45 = 3^2 \cdot 5$ . LCM $= 2^5 \times 3^2 \times 5 = 1440$ .

⚡ Fractions: numerators ↔ denominators cross rule

HCF of fractions = HCF of tops over LCM of bottoms; LCM is the mirror image.

Example: Find the HCF of $\frac{2}{3}$ , $\frac{8}{9}$ and $\frac{10}{27}$ .

$\frac{\gcd(2, 8, 10)}{\text{LCM}(3, 9, 27)} = \frac{2}{27}$ .

⚡ Decimals: shift the point

Multiply everything by the power of 10 needed to clear decimals, solve as integers, shift the point back.

Example: Find the HCF of 0.54, 1.8 and 7.2.

As 54, 180, 720 the HCF is 18 ⇒ answer 0.18.

Where students lose marks

Taking the HCF of fractions as $\frac{\text{HCF of tops}}{\text{HCF of bottoms}}$ .
Forgetting to reduce to lowest terms before checking common primes.
Mixing up min/max powers: highest power goes with LCM.
Returning an integer answer for a decimals question (lost scale factor).

Practice sets — 12 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.