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Factors, multiples and divisibility rules
Rules for 2, 3, 4, 5, 6, 8, 9 and 10, and prime numbers up to 100.
Every whole number is built from smaller numbers multiplied together. Knowing how they fit together makes lots of 11+ questions quick.
In this lesson you will learn:
- what factors and multiples are, and how to tell them apart
- how to find all the factors of a number, using factor pairs
- quick divisibility rules for 2, 3, 4, 5, 6, 8, 9 and 10
- what prime numbers are, and how to find all of them up to 100
1. Factors and multiples
- 3 is a factor of 12, because 12 ÷ 3 = 4 exactly.
- 12 is a multiple of 3, because 3 × 4 = 12.
- 5 is not a factor of 12: 12 ÷ 5 = 2 remainder 2.
Factors come in factor pairs: two numbers that multiply to make the number. 3 × 4 = 12, so 3 and 4 are a factor pair of 12.
The rectangle builder: can the tiles make a rectangle?
- Try every row size for 12 tiles. Which ones make a rectangle?
- Find a number of tiles that only makes one long line. What is special about it?
- Which number up to 48 makes the most rectangles?
2. Finding all the factors
How to find all the factors
- Start with 1 and the number itself1 × 24 = 24
- Try 2, then 3, then 4, and so onIf it divides exactly, write down the pair. If not, skip it.
- Stop when the next number is already in your list
Example: the factors of 24
- 1 × 24, 2 × 12, 3 × 8, 4 × 6
- 5 does not go into 24. The next number, 6, is already in the list, so stop.
The arcs join each factor pair, like a rainbow. If one number pairs with itself (6 × 6 = 36), it only goes in the list once.
The factor rainbow: find the pairs one at a time
3. Multiples
- Every number is a multiple of itself: 7 × 1 = 7.
- Multiples of a number make patterns on a hundred square. Try it below.
The multiples square: pick a times table
- Look at the pattern for 5, then for 10. What do you notice about the last digits?
- Tap any number to see if it is a multiple of the table you picked.
- The multiples of 9 make a slanting line. Why might that be?
4. Divisibility rules
| Divisible by | The rule | Example |
|---|---|---|
| 2 | The last digit is even: 0, 2, 4, 6 or 8 | 3,576 |
| 3 | Add the digits. The total is in the 3 times table. | 4,251: 4 + 2 + 5 + 1 = 12 |
| 4 | The last two digits make a number in the 4 times table | 7,316: 16 ÷ 4 = 4 |
| 5 | The last digit is 0 or 5 | 8,125 |
| 6 | It passes the rules for both 2 and 3 | 174: even, and 1 + 7 + 4 = 12 |
| 8 | The last three digits make a number in the 8 times table | 5,128: 128 ÷ 8 = 16 |
| 9 | Add the digits. The total is in the 9 times table. | 3,825: 3 + 8 + 2 + 5 = 18 |
| 10 | The last digit is 0 | 4,610 |
The divisibility detective: tap each card, then check
Is the number divisible by… Tap once for yes (✓), twice for no (✗).
5. Prime numbers
- 7 is prime: its only factors are 1 and 7.
- 8 is not prime: its factors are 1, 2, 4 and 8.
- 1 is not prime. It has only one factor.
- 2 is the only even prime. Every other even number has 2 as a factor too.
There are 25 prime numbers up to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
A number that is not prime (and not 1) has more than two factors. To check a number up to 100, try dividing by 2, 3, 5 and 7. If none of them goes in exactly, it is prime.
The prime sieve: cross out the multiples, step by step
The circled numbers are the 25 primes up to 100.
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Mixing up factors and multiples. remember factors are few and fit inside; multiples are many and go on forever.
- Missing a factor. work in pairs from 1 upwards, and only stop when the pairs meet.
- Forgetting 1 and the number itself. Every number has both as factors. start every list with 1 × the number.
- Saying 1 is prime. remember a prime has exactly two factors. 1 has only one.
- Thinking every odd number is prime. 9, 15, 21 and 27 are odd but not prime. check by dividing by 3, 5 and 7.
- Using only one rule for 6. 15 is not divisible by 6. it must pass the rules for both 2 and 3.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 1.5.
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