Finding HCF & LCM (incl. fractions and decimals) — notes, question types & practice · HCF & LCM

01

Overview

Every HCF or LCM question starts by picking a method. Small numbers yield to prime factorisation, two big numbers to long division, fractions to the cross rule, and decimals to shifting the point. For 24, 36 and 50: $24 = 2^3 \times 3$ , $36 = 2^2 \times 3^2$ , $50 = 2 \times 5^2$ , so the HCF is $2$ and the LCM is $2^3 \times 3^2 \times 5^2 = 1800$ .

02

Prime factorisation

Write each number as a product of primes, one row per number in a small table.

HCF: multiply the primes common to every number, each at its lowest power.
LCM: multiply every prime that appears, each at its highest power.

Reading the table column by column does the work: column minima give the HCF, column maxima give the LCM. If no prime is common to all numbers, the HCF is 1.

Rule: HCF takes the lowest powers of common primes; LCM takes the highest powers of all primes.

03

Long division for two big numbers

Divide the bigger number by the smaller. Then divide the old divisor by the remainder. Repeat until the remainder is 0; the last non-zero divisor is the HCF. For 432 and 180: $432 = 2 \times 180 + 72$ , then $180 = 2 \times 72 + 36$ , then $72 = 2 \times 36 + 0$ . The HCF is 36. The LCM then falls out as $\dfrac{432 \times 180}{36} = 2160$ .

For three numbers, chain the process: HCF of the first two, then HCF of that result with the third.

Tip: After the HCF, get the LCM as product divided by HCF. Never multiply the numbers out fully first; cancel the HCF first.

04

Numbers already in prime-power form

When the question hands you numbers like $2^4 \times 3 \times 5$ and $2^2 \times 3^3 \times 5^2$ , skip factorising. HCF: lowest power of each prime, here $2^2 \times 3 \times 5 = 60$ . LCM: highest power of each prime, here $2^4 \times 3^3 \times 5^2 = 10800$ . Multiply out only at the end.

05

Fractions: the cross rule

\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}

The two rules mirror each other, so swap them deliberately. HCF of $\dfrac{2}{3}, \dfrac{8}{9}, \dfrac{10}{27}$ is $\dfrac{\gcd(2, 8, 10)}{\text{LCM}(3, 9, 27)} = \dfrac{2}{27}$ .

Watch: Reduce every fraction to lowest terms before applying the cross rule; a reducible fraction corrupts both lists.

06

Decimals: shift the point

Count the largest number of decimal places, multiply everything by that power of 10, solve as whole numbers, then shift the point back. LCM of $0.15, 0.20, 0.45$ : work with $15, 20, 45$ , whose LCM is 180, so the answer is 1.80.

Example: Check by dividing: $1.80 \div 0.15 = 12$ , $1.80 \div 0.20 = 9$ , $1.80 \div 0.45 = 4$ . All whole, so 1.80 is right.

07

Choosing a method fast

Small numbers: factorise mentally. Two big numbers: long division. Prime-power form: read minima and maxima. Fractions: cross rule. Decimals: clear the point, restore it at the end.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1common3 practice Q

HCF by successive (long) division

How to spot it:

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

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

Method

Divide the bigger number by the smaller and note the remainder.
Divide the previous divisor by that remainder.
Repeat until the remainder is 0.
The last non-zero divisor is the HCF; for a third number, chain once more.

Why it works:

Every common divisor of the two numbers also divides each remainder, so the process keeps the common divisors while the numbers shrink.

Try this

Find the HCF of 1617 and 903 by successive division.

Show solution

$1617 = 1 \times 903 + 714$ .
$903 = 1 \times 714 + 189$ , then $714 = 3 \times 189 + 147$ .
$189 = 1 \times 147 + 42$ , then $147 = 3 \times 42 + 21$ , then $42 = 2 \times 21 + 0$ .
HCF $= 21$ .

Answer

21

Type 2common3 practice Q

HCF and LCM from prime-power form

How to spot it:

The numbers arrive as prime products like 2 cubed times 3 squared times 5, and HCF or LCM is asked.

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

Method

List every prime that appears in any number.
HCF: take the smallest exponent of each prime across all numbers; skip a prime missing from any number.
LCM: take the largest exponent of each prime anywhere.
Multiply out only at the end.

Why it works:

The HCF must fit inside every number; the LCM must contain every number.

Try this

Find the LCM of the numbers $2^3 \times 3^2 \times 5$ , $2^2 \times 3^3 \times 7$ and $2 \times 3^2 \times 5^2$ .

Show solution

Highest power of each prime: $2^3$ , $3^3$ , $5^2$ , $7$ .
LCM $= 8 \times 27 \times 25 \times 7$ .
$= 37800$ .

Answer

37800

Type 3very common2 practice Q

HCF and LCM of fractions (cross rule)

How to spot it:

A list of fractions like 2/3, 8/9, 10/27 is given and the HCF or LCM of the fractions is asked.

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

Method

Reduce each fraction to lowest terms.
For the HCF: HCF of the tops over LCM of the bottoms.
For the LCM: LCM of the tops over HCF of the bottoms.
Check: every fraction divided by the HCF is whole; the LCM divided by every fraction is whole.

Why it works:

Putting all fractions over the LCM of the denominators turns them into whole numbers, and the ordinary rules on those integers give the cross rule.

Try this

Find the LCM of 2/3, 8/9 and 10/27.

Show solution

LCM of tops: $\text{LCM}(2, 8, 10) = 40$ .
HCF of bottoms: $\gcd(3, 9, 27) = 3$ .
LCM $= \dfrac{40}{3}$ . Check: $\dfrac{40}{3} \div \dfrac{8}{9} = 15$ , whole.

Answer

40/3

Type 4common2 practice Q

HCF and LCM of decimals

How to spot it:

Decimal values like 0.63, 1.05 and 2.1 are given, and the answer carries a decimal point.

\text{scale by } 10^k \to \text{integer HCF or LCM} \to \text{scale back by } 10^k

Method

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 it works:

A common scale factor multiplies both HCF and LCM by itself, so undo it at the end.

Try this

Find the HCF of 0.63, 1.05 and 2.1.

Show solution

Two decimal places: work with $63, 105, 210$ .
$\gcd(63, 105, 210) = 21$ .
Shift back two places: HCF $= 0.21$ .

Answer

0.21

Type 5common

LCM of two large numbers via the HCF

How to spot it:

The LCM of two numbers too large to factorise is asked directly, or the HCF was already found.

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

Method

Find the HCF first, by long division.
Divide one number by the HCF.
Multiply the quotient by the other number.
Never multiply the two big numbers first; cancel the HCF before multiplying.

Why it works:

The product of two numbers equals HCF times LCM, so dividing by the HCF leaves the LCM without a huge multiplication.

Try this

Find the LCM of 1617 and 903.

Show solution

HCF by division: $\gcd(1617, 903) = 21$ .
Cancel first: $1617 \div 21 = 77$ .
LCM $= 77 \times 903 = 69531$ .

Answer

69531

09

Formula sheet

HCF by factors

\text{HCF} = p_1^{\min} \times p_2^{\min} \times \cdots

common primes, lowest powers

LCM by factors

\text{LCM} = p_1^{\max} \times p_2^{\max} \times \cdots

all primes, highest powers

HCF of fractions

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

LCM of fractions

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

LCM from the HCF

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

10

Shortcuts that save time

⚡ Row of prime powers

Write each number as a row of prime powers in a small grid. LCM reads the column maxima, HCF the common minima. Three numbers take under thirty seconds.

Example

Find the LCM of 18, 24 and 30.

Show solution

$18 = 2 \times 3^2$ , $24 = 2^3 \times 3$ , $30 = 2 \times 3 \times 5$ .
Column maxima: $2^3$ , $3^2$ , $5$ .
LCM $= 8 \times 9 \times 5 = 360$ .

Answer

360

⚡ Cross rule for fractions

HCF of fractions: HCF of tops over LCM of bottoms. LCM of fractions: LCM of tops over HCF of bottoms.

Example

Find the HCF of 3/4, 5/6 and 7/8.

Show solution

HCF of tops: $\gcd(3, 5, 7) = 1$ .
LCM of bottoms: $\text{LCM}(4, 6, 8) = 24$ .
HCF $= \dfrac{1}{24}$ .

Answer

1/24

⚡ Shift the decimal point

Multiply by the power of 10 that clears every decimal, solve as integers, then shift the point back by the same number of places.

Example

Find the LCM of 0.2, 0.25 and 0.4.

Show solution

Two decimal places: work with $20, 25, 40$ .
$\text{LCM}(20, 25, 40) = 200$ .
Shift back two places: $2.00$ .

Answer

2

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Taking the HCF of fractions as HCF of tops over HCF of bottoms.

The denominator of the HCF is the LCM of the bottoms.

Mistake 02

Skipping the reduction of fractions before the cross rule.

Reduce every fraction to lowest terms first.

Mistake 03

Giving the LCM the lowest powers and the HCF the highest.

HCF reads minima, LCM reads maxima.

Mistake 04

Returning a whole-number answer for a decimals question.

The scale factor of 10 must be divided back out at the end.

Mistake 05

Stopping long division at the first zero remainder divisor shown.

The HCF is the last non-zero divisor, not any earlier divisor.

12

Quick revision

Read this the night before the exam.

HCF: common primes, lowest powers. LCM: every prime, highest powers.
Long division: last non-zero divisor is the HCF; LCM $= \dfrac{ab}{\text{HCF}}$ .
Three or more numbers: chain the HCF pairwise.
Prime-power form: read minima and maxima straight off.
Fractions: HCF of tops over LCM of bottoms, and the mirror for LCM.
Decimals: clear the point, solve, restore the same number of places.

13

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

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