Factors and Multiples

Find factors by pairing them up, pick factors and multiples from a list, find common factors and multiples, and find the largest odd factor of a number.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 55 real past-paper questions below

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What Factors and Multiples Are

A factor divides into a number exactly, with nothing left over. A multiple is in the number's times table.

Worked example

1
$factors of 12 : 1, 2, 3, 4, 6, 12$ — Each one divides 12 exactly, with nothing left over: for example $12 \div 4 = 3$ .
2
$multiples of 12 : 12, 24, 36, 48$ — These are $12 \times 1, 12 \times 2, 12 \times 3, 12 \times 4$ : the 12 times table. It carries on for ever.

Rule

Factors are small: never bigger than the number. Multiples are big: never smaller. A number has a few factors, but its multiples go on forever.

Key idea

1 and the number itself are always factors. The number itself is also its first multiple.

Key idea

Add up the factors of 220, leaving out 220 itself, and you get 284. Do the same for 284 and you get 220! Pairs like this are called amicable (friendly) numbers. Thabit ibn Qurra, working in Baghdad in the 800s, found a rule for making them.

Listing All the Factors

The 1-mark question: write down all the factors of a number, like 36. Find them in pairs that multiply to 36, so none go missing.

Worked example

1
$1 \times 36$ — Always start with 1. One times the number itself is always the first pair.
2
$2 \times 18$ — 36 is even, so 2 divides in: $36 \div 2 = 18$ . That gives the next pair.
3
$3 \times 12$ — $36 \div 3 = 12$ with nothing left over, so 3 and 12 are a pair.
4
$4 \times 9$ — $36 \div 4 = 9$ , so 4 and 9 are a pair.
5
$36 \div 5 = 7.2$ — Not a whole number, so 5 is not a factor. Skip it and try 6.
6
$6 \times 6$ — The two numbers in the pair meet, so stop. Every factor has now been found.
7
$1, 2, 3, 4, 6, 9, 12, 18, 36$ — Write the left numbers going up, then the right numbers coming back down. 6 is written once.

Common mistake

Writing 6 twice, or forgetting 1 and 36. The number 36 has nine factors, and 6 counts once.

Key idea

Once the two numbers in a pair meet or cross, you have found every factor. For 36 that happens at 6, because $6 \times 6 = 36$ .

Factors From a List, and Common Factors

The 1 or 2-mark question: from a list, write down the factors of a number. Test each one: does it divide in exactly?

Worked example

1
$7, 9, 14, 16, 25, 30$ — The question: which of these are factors of 28? Divide 28 by each one in turn.
2
$28 \div 7 = 4$ — Yes: no remainder, so 7 is a factor.
3
$28 \div 9 = 3.1 \dots$ — Not a whole number, so 9 is not a factor.
4
$28 \div 14 = 2$ — Yes: no remainder, so 14 is a factor.
5
$28 \div 16 = 1.75$ — Not a whole number, so 16 is not a factor.
6
$28 \div 25 = 1.12$ — Not a whole number, so 25 is not a factor.
7
$30 > 28$ — 30 is bigger than 28, so it cannot be a factor.
8
$7, 14$ — The answer. If the question asks for just a factor, either one scores.

A common factor of 24 and 40 divides into both of them.

Worked example

1
$24 : 1, 2, 3, 4, 6, 8, 12, 24$ — List the factors of 24 in pairs, as in the last card.
2
$40 : 1, 2, 4, 5, 8, 10, 20, 40$ — List the factors of 40 the same way. The blue numbers appear in both lists.
3
$1, 2, 4, 8$ — These divide both 24 and 40, so they are the common factors.

Common mistake

Giving 30 as a factor of 28 because it is close. A factor can never be bigger than the number.

Key idea

"A common factor" means any one of them: 1, 2, 4 or 8. The biggest is the HCF, in the lesson HCF and LCM.

Quick Tests for Factors

No calculator, and a big number? These quick tests tell you if a small number is a factor, just by looking at the digits.

To divide by	Check this
2	the last digit is even
3	the digits add up to a multiple of 3
4	the last two digits make a multiple of 4
5	the last digit is 0 or 5
6	it passes the tests for 2 and 3
9	the digits add up to a multiple of 9
10	the last digit is 0
11	take away and add the digits in turn; the answer is 0 or a multiple of 11

Worked example

1
$7362$ — The question: which of 2, 3, 4, 5 and 9 are factors? Test each one with its rule.
2
$7362$ — The last digit, 2, is even, so 2 is a factor.
3
$7 + 3 + 6 + 2 = 18$ — Add the digits. The tests for 3 and 9 both use this digit sum.
4
$18 \div 3 = 6, 18 \div 9 = 2$ — 18 divides by 3 and by 9, so 7 362 does too. Both are factors.
5
$62 \div 4 = 15.5$ — For 4, look at the last two digits only. 62 is not in the 4 times table, so 4 is not a factor.
6
$last digit 2$ — It is not 0 or 5, so 5 is not a factor.
7
$2, 3, 9$ — The answer: the numbers that passed.

The 2-mark “show that” question uses the test for 11. Show that 8 254 917 is a multiple of 11.

Worked example

1
$8 - 2 + 5 - 4 + 9 - 1 + 7$ — Start with the first digit, then take away and add the digits in turn.
2
$8 + 5 + 9 + 7 = 29$ — Add the plus digits: the 1st, 3rd, 5th and 7th.
3
$2 + 4 + 1 = 7$ — Add the minus digits: the 2nd, 4th and 6th.
4
$29 - 7 = 22$ — Take the second total from the first.
5
$22 = 2 \times 11$ — 22 is a multiple of 11, so 8 254 917 is a multiple of 11. Write this sentence: the mark is for the reason.

Common mistake

Using the digit sum to test for 4 or 8. Adding the digits only works for 3 and 9.

Writing Down a Multiple

The 1-mark question: write down a multiple of 8 between 30 and 60. Go up the 8 times table and stop inside the range.

Worked example

1
$8, 16, 24, 32$ — Go up the 8 times table. 32 is the first one past 30.
2
$32, 40, 48, 56$ — Keep adding 8. All four are between 30 and 60, so any one scores. The next, 64, is too big.

A multiple of both 6 and 9 must be in both times tables.

Worked example

1
$6 : 6, 12, 18, 24, 30, 36$ — Write the 6 times table first.
2
$9 : 9, 18, 27, 36$ — Then the 9 times table. 18 and 36 are in both tables.
3
$18$ — So 18 is a common multiple of 6 and 9. It is the smallest one, and 36 also works.

Common mistake

Giving a factor, like 4 or 2, when the question asks for a multiple of 8. A multiple is 8 or bigger.

Key idea

Multiplying the two numbers always works: $6 \times 9 = 54$ is a multiple of both. It is just not always the smallest.

Factor of One, Multiple of Another

The 3-mark question: a number is a factor of one number and a multiple of another. Find all the values. List the factors, then keep the ones that pass the second test.

Find every n that is a factor of 120 and a multiple of 6.

Worked example

1
$1 \times 120, 2 \times 60, 3 \times 40, 4 \times 30$ — Find the factors of 120 in pairs, starting with 1.
2
$5 \times 24, 6 \times 20, 8 \times 15, 10 \times 12$ — Keep going. 7 and 9 do not divide in. After 10 × 12, 11 fails and 12 is already found, so stop.
3
$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$ — All 16 factors in order. Now test each one: is it in the 6 times table?
4
$6, 12, 24, 30, 60, 120$ — These divide by 6 exactly, so they are the answers. 6 counts, because $6 = 6 \times 1$ .

Key idea

Show your list of factors. Even if you miss one answer, the list can still earn a mark.

Common mistake

Leaving out 6 or 120. Every number is a multiple of itself and a factor of itself.

The Largest Odd Factor

The harder 1-mark question: find the largest factor of a number that is odd. Take out every factor of 2, and what is left is the answer.

Worked example

1
$720 \div 2 = 360$ — 720 ends in 0, so it is even. Halve it to take out one factor of 2.
2
$360 \div 2 = 180$ — 360 is still even, so halve again to take out another 2.
3
$180 \div 2 = 90$ — 180 is even too. Halve again.
4
$90 \div 2 = 45$ — 45 ends in 5, so it is odd: no 2s are left. Stop here.
5
$720 = 2^{4} \times 45$ — Check: $16 \times 45 = 720$ . Every odd factor of 720 divides 45, so 45 is the largest.

Key idea

If the number is given as primes, just drop the 2s: $2^{4} \times 3^{2} \times 5$ gives $3^{2} \times 5 = 45$ .

Common mistake

Answering a small odd factor like 3 or 5. They are odd factors, but not the largest.

Largest Odd Common Factor

The 1 or 2-mark question: find the largest odd number that is a factor of two numbers. Find the shared primes, then take out every 2.

Find the largest odd number that is a factor of 84 and of 126.

Worked example

1
$84 = 2 \times 2 \times 3 \times 7$ — Write 84 as a product of primes, as in Prime Factorization.
2
$126 = 2 \times 3 \times 3 \times 7$ — Do the same for 126.
3
$2 \times 3 \times 7 = 42$ — These primes are in both lists. Their product, 42, is the HCF, but it is even.
4
$3 \times 7$ — An odd number has no factor 2, so drop the 2 and keep the rest.
5
$3 \times 7 = 21$ — Multiply what is left. Check: $84 \div 21 = 4$ and $126 \div 21 = 6$ , and 21 is odd.

Key idea

In the 2-mark version, writing both numbers as primes earns a mark even if the last step goes wrong. Show them.

Common mistake

Giving the HCF, 42. It divides both numbers, but it is even. Answers like 3 or 7 are odd, but not the largest.

Which Method? Read the Question

Find the key words, then pick the method.

You see	You do
All the factors	find them in pairs
A factor of, from a list	test: does it divide exactly?
A multiple of	go up the times table
Common factor	a number that divides both
Multiple of both	in both times tables
Largest odd factor	halve until it is odd
Is it a factor of a big number?	use the digit tests
A factor of one number and a multiple of another	list the factors, keep the multiples
Largest odd common factor	shared primes, then drop the 2s
Highest common factor or lowest common multiple	HCF and LCM
A prime number	Prime Numbers
Product of prime factors	Prime Factorization

Key idea

Check the size: a factor is never bigger than the number, and a multiple is never smaller.

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