HCF and LCM

Find the highest common factor and lowest common multiple of numbers, from prime factors or from powers, and use them in timing and packing problems.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 91 real past-paper questions below

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What HCF and LCM Mean

Two numbers can share factors, and they can share multiples.

Word	Meaning
HCF	the biggest number that divides into both
LCM	the smallest number both divide into

Worked example

1
$factors of 12 : 1, 2, 3, 4, 6, 12$ — Write every number that divides 12 exactly, with nothing left over.
2
$factors of 18 : 1, 2, 3, 6, 9, 18$ — Do the same for 18.
3
$common factors: 1, 2, 3, 6$ — These are in both lists, so they divide both numbers.
4
$HCF = 6$ — The biggest common factor is the highest common factor.
5
$multiples of 12 : 12, 24, 36, 48$ — Go up the 12 times table.
6
$multiples of 18 : 18, 36, 54$ — Go up the 18 times table.
7
$LCM = 36$ — 36 is the first number in both tables, so it is the lowest common multiple.

Rule

The HCF is never bigger than your numbers. The LCM is never smaller.

Key idea

In the USA, some cicadas come out of the ground every 13 years and others every 17 years. The LCM is $13 \times 17 = 221$ , so two such groups appear together only once every 221 years. In 2024, two neighbouring groups did just that.

HCF and LCM From Prime Factors

The 2-mark question: find the HCF or LCM of two numbers like 84 and 120. Listing is too slow. Split each number into primes.

Worked example

1
$84 = 2 \times 2 \times 3 \times 7$ — Split 84 into primes, as in Prime Factorization. Write every prime, one at a time.
2
$120 = 2 \times 2 \times 3 \times 2 \times 5$ — Split 120 the same way. Put first the primes it shares with 84: two 2s and a 3.
3
$HCF = 2 \times 2 \times 3$ — The HCF uses only the shared primes, because only they divide both numbers.
4
$HCF = 12$ — Multiply them: 2 × 2 = 4, and 4 × 3 = 12.
5
$LCM = 12 \times 7 \times 2 \times 5$ — Take the shared part once, then add the leftovers: 7 from 84, and 2 and 5 from 120.
6
$LCM = 84 \times 10 = 840$ — 12 × 7 = 84 and 2 × 5 = 10, then 84 × 10 = 840.
7
$840 \div 84 = 10, 840 \div 120 = 7$ — Check: both numbers divide into 840 exactly, so it is a common multiple.

Common mistake

Giving $84 \times 120 = 10080$ as the LCM. It is a common multiple, but not the lowest. Count the shared primes once only.

Key idea

Show both prime factor lines. They earn a mark for working on their own, even if a later slip spoils the answer.

Numbers Given as Powers

The 2-mark question gives numbers as products of primes, like $A$ and $B$ below. Do not multiply them out. Compare the power of each prime.

A = 2^{3} \times 3^{2} \times 5

B = 2^{2} \times 3 \times 5^{2} \times 7

Worked example

1
$shared primes: 2, 3, 5$ — Both numbers have a 2, a 3 and a 5. The 7 is only in $B$ .
2
$HCF = 2^{2} \times 3 \times 5$ — For each shared prime take the lowest power: $2^{2}$ not $2^{3}$ , $3^{1}$ not $3^{2}$ , $5^{1}$ not $5^{2}$ . That is the most that divides both.
3
$HCF = 4 \times 3 \times 5 = 60$ — Only if the question wants a number: $2^{2} = 4$ , then multiply.
4
$LCM = 2^{3} \times 3^{2} \times 5^{2} \times 7$ — Take the highest power of every prime in either number, 7 included, so that both $A$ and $B$ divide into it.

Common mistake

Putting 7 in the HCF. The 7 is only in $B$ , so it is not shared. It does go in the LCM.

Asked about $2 A$ and $3 B$ ? Write each one in powers first, then use the same rule.

Worked example

1
$2 A = 2^{4} \times 3^{2} \times 5$ — 2 is $2^{1}$ , so the power of 2 goes up by 1: 3 + 1 = 4.
2
$3 B = 2^{2} \times 3^{2} \times 5^{2} \times 7$ — 3 is $3^{1}$ , so the power of 3 goes up by 1: 1 + 1 = 2.
3
$LCM = 2^{4} \times 3^{2} \times 5^{2} \times 7$ — Now take the highest power of each prime, as before.

Three Numbers at Once

Some questions give three numbers. The rules do not change, but for the HCF a prime must be in all three.

Worked example

1
$18 = 2 \times 3^{2}$ — Split each number into primes, with powers: 18 = 2 × 9.
2
$24 = 2^{3} \times 3$ — 24 = 8 × 3.
3
$40 = 2^{3} \times 5$ — 40 = 8 × 5.
4
$in all three: only 2$ — 3 is missing from 40, and 5 is missing from 18 and 24. So only 2 is shared by all three.
5
$HCF = 2^{1} = 2$ — The lowest power of 2 is $2^{1}$ , from 18.
6
$LCM = 2^{3} \times 3^{2} \times 5$ — Take the highest power of every prime that appears anywhere: $2^{3}$ , $3^{2}$ and 5.
7
$LCM = 8 \times 9 \times 5 = 360$ — 8 × 9 = 72, and 72 × 5 = 360.
8
$360 \div 18 = 20, 360 \div 24 = 15, 360 \div 40 = 9$ — Check: all three divide into 360 exactly.

Common mistake

Putting 3 in the HCF because 18 and 24 share it. Every number must share it, and 40 does not.

Key idea

Three numbers already given as powers? Skip the splitting and go straight to the two rules.

LCM Word Questions: Times and Packs

The 3-mark word question: two things repeat at different times. When do they happen together again? That is the LCM.

Buses leave every 15 minutes and every 25 minutes. Both leave at 8:00. When do they next leave together?

Worked example

1
$15 = 3 \times 5$ — Split 15 into primes.
2
$25 = 5 \times 5$ — Split 25 into primes. The two numbers share one 5.
3
$LCM = 5 \times 3 \times 5$ — The shared 5 once, then the leftovers: 3 from 15 and the other 5 from 25.
4
$LCM = 75 minutes$ — 5 × 3 × 5 = 75. The first time both timetables meet is 75 minutes after 8:00.
5
$75 min = 60 + 15 = 1 h 15 min$ — Change to hours: 60 minutes make 1 hour, with 15 minutes left.
6
$8 : 00 + 1 h 15 min = 9 : 15$ — Add it to the start time. The question asks for a time, not minutes.

Common mistake

Stopping at 75. The question asks for a time, so add 75 minutes to 8:00.

Key idea

Write the LCM in minutes clearly. It earns most of the marks on its own; the last mark is for the time.

Same idea for packs. Plates come in packs of 10 for $3.50. Cups come in packs of 8 for $2.40. You want the same number of plates and cups. Find the smallest cost.

Worked example

1
$10 = 2 \times 5$ — Split the pack sizes into primes.
2
$8 = 2^{3}$ — 8 = 2 × 2 × 2.
3
$LCM = 2^{3} \times 5 = 40$ — Highest power of each prime: 8 × 5 = 40. So 40 plates and 40 cups is the smallest equal amount.
4
$40 \div 10 = 4 packs of plates$ — How many packs of 10 make 40.
5
$40 \div 8 = 5 packs of cups$ — How many packs of 8 make 40.
6
$4 \times 3.50 = 14.00$ — Cost of the plates: 4 packs at $3.50.
7
$5 \times 2.40 = 12.00$ — Cost of the cups: 5 packs at $2.40.
8
$14.00 + 12.00 = $26.00$ — Add them for the total cost.

Common mistake

Answering 40. That is the number of plates, not the packs or the cost. Divide by each pack size first.

Biggest Equal Pieces: Use the HCF

When you cut or share things into the biggest equal pieces with nothing left over, use the HCF. Each piece must divide every amount.

Two ribbons are 90 cm and 126 cm long. Cut both into pieces of the same length, as long as possible, with no waste.

Worked example

1
$90 = 2 \times 3^{2} \times 5$ — Split 90 into primes: 90 = 2 × 45, and 45 = 9 × 5.
2
$126 = 2 \times 3^{2} \times 7$ — Split 126 into primes: 126 = 2 × 63, and 63 = 9 × 7.
3
$HCF = 2 \times 3^{2} = 18$ — The shared primes are a 2 and $3^{2}$ . 2 × 9 = 18, so each piece is 18 cm.
4
$90 \div 18 = 5$ — The 90 cm ribbon makes 5 pieces.
5
$126 \div 18 = 7$ — The 126 cm ribbon makes 7 pieces.
6
$5 + 7 = 12 pieces$ — Add them, if the question asks how many pieces in total.

Common mistake

Using the LCM, 630. A piece cannot be longer than the ribbon. For cutting or sharing, the answer is never bigger than your numbers.

Memory trick

Split up into equal groups: HCF. Meet again or match up: LCM.

From HCF and LCM to the Numbers

Some questions give the HCF and the LCM and ask for the two numbers. This fact helps:

Formula

HCF \times LCM = a \times b

$a$ and $b$ are the two numbers. The HCF times the LCM is always the same as the two numbers multiplied. It works for two numbers only.

The HCF of two numbers is 4 and their LCM is 60. Both numbers are bigger than 4. Find them.

Worked example

1
$a \times b = 4 \times 60 = 240$ — Use the formula: the two numbers multiply to 240.
2
$60 \div 4 = 15$ — Divide the LCM by the HCF. Each number is 4 times a piece of this 15.
3
$15 = 1 \times 15 or 3 \times 5$ — Split 15 into two parts with no common factor. If they shared one, the HCF would be bigger than 4.
4
$4 \times 1 = 4, 4 \times 15 = 60$ — The first pair gives 4 and 60. Reject it: the question says both numbers are bigger than 4.
5
$4 \times 3 = 12, 4 \times 5 = 20$ — The second pair gives 12 and 20.
6
$12 \times 20 = 240$ — Check with the formula: 240, the same as HCF × LCM.

Key idea

Check the HCF and LCM too: $12 = 2^{2} \times 3$ and $20 = 2^{2} \times 5$ share $2^{2} = 4$ , and $2^{2} \times 3 \times 5 = 60$ .

Common mistake

Giving 4 and 60. They do have HCF 4 and LCM 60, but they break the condition. Always reread the question for conditions.

Letters in Powers (Extended/Higher)

Higher questions put letters in the powers, or ask about algebra terms. The rules do not change: lowest power for the HCF, highest power for the LCM.

Here $A = 2^{n + 3} \times 3^{n} \times 5$ and $B = 2^{n} \times 3^{n + 2} \times 7$ , where n is a positive whole number.

Worked example

1
$2^{n} vs 2^{n + 3}$ — Compare the powers of 2. n is less than n + 3, so $2^{n}$ is the lower one.
2
$3^{n} vs 3^{n + 2}$ — Compare the powers of 3. n is less than n + 2, so $3^{n}$ is the lower one.
3
$HCF = 2^{n} \times 3^{n}$ — Lowest power of each shared prime. 5 and 7 are not shared.
4
$LCM = 2^{n + 3} \times 3^{n + 2} \times 5 \times 7$ — Highest power of each prime, plus the 5 and the 7.

A missing power. $N = 2^{x} \times 3 \times 5$ . The LCM of N and 72 is 720. Find $x$ .

Worked example

1
$72 = 2^{3} \times 3^{2}$ — Split 72 into primes: 72 = 8 × 9.
2
$720 = 2^{4} \times 3^{2} \times 5$ — Split the LCM into primes: 720 = 16 × 45.
3
$highest of x and 3 = 4$ — The LCM's power of 2 is the highest of N's power and 72's power.
4
$x = 4$ — 72 only has $2^{3}$ , so the $2^{4}$ must come from N.

Algebra terms. Find the LCM and HCF of $6 x^{3} y$ and $4 x y^{2}$ .

Worked example

1
$6 = 2 \times 3, 4 = 2^{2}$ — Do the numbers first, the same as always.
2
$2^{2} \times 3 = 12$ — LCM of the numbers: highest power of each prime.
3
$x^{3}, y^{2}$ — Highest power of each letter: $x^{3}$ beats $x$ , and $y^{2}$ beats $y$ .
4
$LCM = 12 x^{3} y^{2}$ — Put the number and the letters together.
5
$HCF = 2 x y$ — For the HCF take the lowest: 2 from the numbers, then $x$ and $y$ .

Common mistake

Adding the powers, like $x^{3} \times x = x^{4}$ . For the LCM you choose the highest power. You do not multiply the terms.

Key idea

If the question gives conditions like $m > 2$ , use them to decide which power is lower.

Which Method? Read the Question

Find the key words, then pick HCF or LCM.

You see	You do
Highest common factor	shared primes only
Lowest common multiple	shared primes once, plus the rest
Numbers written as powers	HCF: lowest powers. LCM: highest powers
Next time together, same number of	LCM, then answer the question
Three numbers	HCF: a prime must be in all three
Same number of each, packs	LCM, then divide by each pack size
Biggest equal pieces or groups	HCF
HCF and LCM given, find the numbers	HCF × LCM = the two numbers multiplied
Letters in the powers (Extended/Higher)	compare: n is less than n + 3
Split a number into primes	Prime Factorization
List factors or multiples	Factors and Multiples

Key idea

Check your answer: the HCF must divide into both numbers, and both numbers must divide into the LCM.

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