Factors, Multiples and Primes
Learn how to find factors and multiples, recognise primes and use simple divisibility checks. Start with equal groups and build towards GCSE problems.
Foundation skills for both tiers. An optional challenge connects square numbers with factor pairs.
Start with equal groups
If 12 counters can be split into groups of 3 with none left over, 3 is a factor of 12.
If you keep making groups of 3, the totals 3, 6, 9, 12, … are multiples of 3.
We will use equal groups to understand factors, multiples and primes. We use positive whole-number factors throughout.
Example Question:
You have 12 counters. Can you arrange them in rows of 5 with none left over?
No: 5 is not a factor of 12. Example 1 finds all the row sizes that do work.
Exam Tip
Look for the word factor or multiple before you start listing.
A factor divides into the number. A multiple is a result in its multiplication table.
Method — Search without missing numbers
Factors: start at 1 and test whole numbers in order. Record each successful pair.
Multiples: start with one group and keep adding the original positive number.
Common: compare two lists and select entries found in both.
Prime: check whether there are exactly two positive factors. Divisibility rules can help.
Worked Examples
Example 1 — Build factor pairs
A factor divides exactly into a number with no remainder. We are looking for positive whole-number factors in this lesson.
Find the factors of 12. A factor pair is two whole numbers that multiply to make 12.
| Factor pair | Equal groups |
|---|---|
| 1 × 12 = 12 | 1 group of 12 |
| 2 × 6 = 12 | 2 groups of 6 |
| 3 × 4 = 12 | 3 groups of 4 |
At 4 × 3 the pair repeats in the other order. We already have both factors.
Put each factor in the list once: 1, 2, 3, 4, 6, 12.
1 is a factor of every positive whole number. The number itself is also a factor.
Example 2 — Search systematically
Find all positive factors of 20. Start testing at 1.
| Test | Result |
|---|---|
| 20 ÷ 1 = 20 | Pair: 1 and 20 |
| 20 ÷ 2 = 10 | Pair: 2 and 10 |
| 20 ÷ 3 is not a whole number | 3 is not a factor |
| 20 ÷ 4 = 5 | Pair: 4 and 5 |
The next candidate is 5, which would pair with 4. We have reached the pairs in reverse order.
The complete list is 1, 2, 4, 5, 10, 20.
Keep testing in order until the candidate has reached or passed its partner. A failed test does not mean you should stop.
Example 3 — List positive multiples
A multiple is a number made by multiplying by a whole number. Here we list positive multiples, starting with one group.
Find the first five positive multiples of 4.
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
Answer: 4, 8, 12, 16, 20. Add another 4 to keep going.
The dots in 4, 8, 12, … mean the list continues. There is no largest positive multiple.
Zero is also a multiple because 4 × 0 = 0, but it is not a positive multiple.
Example 4 — Find common factors
Common means shared. Find the common factors of 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12.
Factors of 18: 1, 2, 3, 6, 9, 18.
Choose the numbers in both lists: 1, 2, 3, 6.
Each answer must divide both original numbers exactly. For example, 12 ÷ 3 = 4 and 18 ÷ 3 = 6.
Example 5 — Find common multiples
Find the first two positive common multiples of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24, …
Multiples of 6: 6, 12, 18, 24, …
The first two numbers in both lists are 12 and 24.
Check: 12 = 4 × 3 = 6 × 2. Also, 24 = 4 × 6 = 6 × 4.
Common multiples are shared results in the tables. Common factors divide both numbers.
Example 6 — Decide whether a number is prime
A prime number is a whole number greater than 1 with exactly two positive factors: 1 and itself.
| Number | Positive factors | Classification |
|---|---|---|
| 2 | 1, 2 | Prime: exactly two |
| 5 | 1, 5 | Prime: exactly two |
| 6 | 1, 2, 3, 6 | Composite: more than two |
| 1 | 1 | Neither prime nor composite |
A composite number is a whole number greater than 1 that is not prime. It has factors other than 1 and itself.
1 is not prime: its only positive factor is 1. Writing 1 twice does not create a second factor.
2 is the only even prime. Any larger even number has 2 as an extra factor.
Odd does not mean prime. For example, 9 = 3 × 3, so 9 is composite.
Example 7 — Test a less familiar number
Is 29 prime? Look for a factor pair other than 1 × 29.
2 does not divide 29: it is odd.
3 does not divide 29: 3 × 9 = 27 and 3 × 10 = 30.
4 does not divide 29: 4 × 7 = 28.
5 does not divide 29: 5 × 5 = 25 and 5 × 6 = 30.
We can stop before 6 because 6 × 6 = 36, which is greater than 29. A pair with both factors at least 6 would be too large.
Any other pair would therefore have included a smaller factor we have already tested.
Only 1 and 29 remain, so 29 is prime.
| Useful primes up to 50 |
|---|
| 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 |
Use this list to build familiarity. When unsure, check factors rather than guessing from the final digit.
Example 8 — Use divisibility rules
A divisibility rule is a quick check for exact division. A digit sum means adding the digits.
Test 246 for divisibility by 2, 3 and 6.
The final digit is 6, which is even. So 246 is divisible by 2.
The digit sum is 2 + 4 + 6 = 12. Since 12 is divisible by 3, 246 is divisible by 3.
It passes both checks, so 246 is divisible by 6. Indeed, 246 ÷ 6 = 41.
Passing just one check is not enough for divisibility by 6.
| Divisor | Quick check | Example |
|---|---|---|
| 2 | Final digit is 0, 2, 4, 6 or 8 | 246 ends in 6 |
| 3 | Digit sum is divisible by 3 | 246: 2 + 4 + 6 = 12 |
| 4 | Final two digits form a number divisible by 4 | 316: 16 ÷ 4 = 4 |
| 5 | Final digit is 0 or 5 | 235 ends in 5 |
| 6 | Divisible by both 2 and 3 | 246 passes both |
| 8 | Final three digits form a number divisible by 8 | 1128: 128 ÷ 8 = 16 |
| 9 | Digit sum is divisible by 9 | 729: 7 + 2 + 9 = 18 |
| 10 | Final digit is 0 | 450 ends in 0 |
For a number with fewer than two or three digits, use the whole number in the 4 or 8 check.
The final-digit rule for 2 works because every full group of ten is even. Only the units can change whether the total is even.
Example 9 — Choose and combine the checks
Is 372 divisible by 4 and by 9?
For 4, use the last two digits: 72 ÷ 4 = 18. So yes; 372 ÷ 4 = 93.
For 9, add the digits: 3 + 7 + 2 = 12. Twelve is not divisible by 9, so no.
Being divisible by 3 does not automatically make a number divisible by 9.
Is 1128 divisible by 8?
Use the final three digits: 128 ÷ 8 = 16. So yes; 1128 ÷ 8 = 141.
Example 10 — Arrange objects in equal rows
There are 18 chairs. Each row must contain the same number, with none left over. What row sizes are possible?
We need factors of 18, because each row size must divide 18 exactly.
Factor pairs: 1 × 18, 2 × 9, 3 × 6.
Possible chairs per row: 1, 2, 3, 6, 9 or 18.
For example, 3 rows of 6 use all 18 chairs.
| Rows | Chairs per row | Total |
|---|---|---|
| 1 | 18 | 18 |
| 2 | 9 | 18 |
| 3 | 6 | 18 |
| 6 | 3 | 18 |
| 9 | 2 | 18 |
| 18 | 1 | 18 |
If the question required more than one row and more than one chair in each row, exclude the first and last arrangements.
Example 11 — Notice what happens for a square — challenge
A square number is a whole number multiplied by itself. For example, 6 × 6 = 36.
Find the factor pairs of 36:
1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
The positive factors are 1, 2, 3, 4, 6, 9, 12, 18, 36: nine factors.
Most pairs contribute two different factors. The central pair contributes 6 just once.
That is why a positive square number has an odd number of positive factors. This is an optional challenge connection, useful at both tiers.
Key ideas to remember
| Idea | Meaning |
|---|---|
| Factor | Divides a number exactly |
| Multiple | A result in a multiplication table |
| Common | Shared by both lists |
| Prime | Exactly two positive factors |
| Composite | A whole number greater than 1 that is not prime |
A positive whole number has a finite list of positive factors. Its positive multiples continue indefinitely.
1 is neither prime nor composite. 2 is the only even prime.
Check your answer
Divide the original number by every proposed factor. Each answer must be a whole number.
Multiply each factor pair to check that it makes the original number.
Check that common factors divide both numbers, and common multiples appear in both tables.
A positive factor cannot exceed the original positive number. A positive multiple cannot be smaller than it.
Do not call an odd number prime until you have checked for other factors.
Think Like a Mathematician
Why is 2 the only even prime?
An even number can be divided exactly by 2.
For an even number greater than 2, the factors include 1, 2 and the number itself. That is already more than two.
For 2, those are just 1 and 2. There is no extra factor, so 2 is prime.
Common mistakes to avoid
Swapping factors and multiples: 3 is a factor of 12; 12 is a multiple of 3.
Missing the ends: include 1 and the original number when listing positive factors.
Stopping at the first failed test: 3 does not divide 20, but 4 does. Test systematically.
Calling 1 prime: it has only one positive factor, not two.
Assuming odd means prime: 9 and 15 are odd composite numbers.
Rejecting 2 because it is even: count its factors; 1 and 2 are exactly two.
Mixing divisibility rules: use the last two digits for 4, but the digit sum for 3 or 9.
Confusing common factors with common multiples: decide whether the answer divides both numbers or is a multiple of both.
Try it yourself
Begin with small lists. Show your pairs or divisibility checks before opening the worked answer.
1. Write the factor pairs of 12 and list all its positive factors.
1 × 12, 2 × 6 and 3 × 4 each make 12.
After this the pairs reverse. List each number once: 1, 2, 3, 4, 6, 12.
2. Find all positive factors of 16.
Test in order: 1 × 16, 2 × 8. Three is not a factor. Then 4 × 4 = 16.
The repeated central factor is listed once.
Answer: 1, 2, 4, 8, 16.
3. Write the first five positive multiples of 7.
Multiply 7 by 1, 2, 3, 4 and 5.
Answer: 7, 14, 21, 28, 35. The list continues by adding 7.
4. Is 8 a factor of 24? Is 24 a multiple of 8?
24 ÷ 8 = 3 with no remainder, so 8 is a factor of 24.
8 × 3 = 24, so 24 is a multiple of 8. Both statements are true.
5. Find the common factors of 16 and 24.
Factors of 16: 1, 2, 4, 8, 16.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Choose the shared entries: 1, 2, 4, 8.
6. Find the first two positive common multiples of 3 and 5.
Multiples of 3 up to 30: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
Multiples of 5 up to 30: 5, 10, 15, 20, 25, 30.
The first two shared positive entries are 15 and 30.
7. Classify 1, 2, 9, 11 and 15 as prime, composite or neither.
1 has one positive factor: neither.
2 has factors 1 and 2: prime.
9 has factors 1, 3 and 9: composite.
11 has only factors 1 and 11: prime.
15 has factors 1, 3, 5 and 15: composite.
8. Is 39 prime? Explain.
3 + 9 = 12, which is divisible by 3.
39 = 3 × 13. It has factors other than 1 and itself.
So 39 is composite.
9. Which of 2, 3, 5, 6, 9 and 10 divide 270 exactly?
270 ends in 0, so it is divisible by 2, 5 and 10.
The digit sum is 2 + 7 + 0 = 9, so it is divisible by 3 and 9.
It passes the 2 and 3 checks, so it is divisible by 6.
All six divide exactly. The quotients are 135, 90, 54, 45, 30 and 27 respectively.
10. Use quick checks to decide whether 524 is divisible by 4 and 2136 is divisible by 8.
524: the final two digits make 24, and 24 ÷ 4 = 6. Yes; 524 ÷ 4 = 131.
2136: the final three digits make 136, and 136 ÷ 8 = 17. Yes; 2136 ÷ 8 = 267.
11. Twenty-four plants go in equal rows, with no plants left over. List all possible numbers of plants per row.
Find factor pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
Five does not divide 24. At 6 × 4, the pairs reverse.
Possible row sizes: 1, 2, 3, 4, 6, 8, 12, 24 plants.
12. One light flashes every 4 seconds and another every 6 seconds. They flash together now. Give the first two positive times when they flash together again.
After now, the first light flashes at 4, 8, 12, 16, 20, 24, … seconds.
The second flashes at 6, 12, 18, 24, … seconds.
The first two shared times are 12 seconds and 24 seconds after now.
13. Challenge: list the positive factors of 25. Why is the number of factors odd?
Factor pairs: 1 × 25 and 5 × 5.
Factors: 1, 5, 25. There are three.
The pair 5 × 5 contributes just one different factor. That leaves an odd total.
Frequently Asked Questions
Is 1 a prime number?
No. It has only one positive factor. A prime must have exactly two.
Are all odd numbers prime?
No. For example, 9 = 3 × 3 and 15 = 3 × 5.
Is zero a multiple?
Yes. Multiplying a number by zero gives zero. Questions asking for positive multiples start above zero.
Can a number be both a factor and a multiple of itself?
Yes. For example, 6 ÷ 6 = 1 and 6 × 1 = 6.
How do I know a factor list is complete?
Test candidates in order. Once the smaller candidate reaches or passes its partner, any remaining pair would reverse one already checked.
Where will I learn prime factorisation, HCF and LCM?
The Prime Factorisation, HCF and LCM lesson develops these methods. Here we have built the factor and multiple ideas they need.