Multiplication and division (1)
Tests of divisibility
A divisibility test lets you quickly find out whether a number can be divided by another with no remainder. In Stage 6 we learn the tests for 3, 6 and 9. They all use the sum of the digits.
Divisible by 3
A number is divisible by 3 if the sum of its digits is a multiple of 3. Example: for 43 719, the digits add to 4 + 3 + 7 + 1 + 9 = 24. Since 24 is a multiple of 3, the number is divisible by 3.
Divisible by 9
A number is divisible by 9 if the sum of its digits is a multiple of 9. Example: 495 has digit sum 4 + 9 + 5 = 18, a multiple of 9, so it is divisible by 9. But 987 has digit sum 24, which is not a multiple of 9, so it is not divisible by 9.
Divisible by 6
Because 6 = 2 × 3, a number is divisible by 6 when it is even AND divisible by 3. If a number is odd it can never be divisible by 6, even when its digits add to a multiple of 3.
Finding missing digits
You can use these tests to fill in a missing digit. Add the digits you know, then choose a digit that makes the total a multiple of 3 (or 9). There can be more than one answer.
Worked examples
Testing for divisibility by 3
- Test whether 43 719 is divisible by 3.
- Add the digits: 4 + 3 + 7 + 1 + 9 = 24.
- 24 is a multiple of 3.
Answer: 43 719 is divisible by 3.
Testing for divisibility by 9
- Test whether 495 is divisible by 9.
- Add the digits: 4 + 9 + 5 = 18.
- 18 is a multiple of 9.
Answer: 495 is divisible by 9.
Testing for divisibility by 6
- Test whether 3594 is divisible by 6.
- Check it is even: it ends in 4, so yes.
- Check it is divisible by 3: 3 + 5 + 9 + 4 = 21, a multiple of 3.
Answer: 3594 is even and divisible by 3, so it is divisible by 6.
Finding a missing digit
- Write a digit in the box so 80□17 is divisible by 3.
- Add the digits you know: 8 + 0 + 1 + 7 = 16.
- Choose a digit that makes the total a multiple of 3: 16 + 2 = 18, 16 + 5 = 21, 16 + 8 = 24.
Answer: The digit can be 2, 5 or 8, giving 80 217, 80 517 or 80 817.
Using divisibility to solve a puzzle
- A number between 200 and 220 is divisible by 6 and its digits add up to 3.
- List the multiples of 6 in that range: 204, 210, 216.
- Check the digit sums: 204 → 6, 210 → 3, 216 → 9.
Answer: The number is 210.
Where you see this in real life
- Sharing cookies equally between yourself and five friends (dividing by 6).
- Quickly checking whether a number of items can be split into equal groups.
- Sorting numbers into a Venn diagram of 'divisible by 3', 'by 6' and 'by 9'.
Key words
- Divisible
- Can be divided by another number with no remainder.
- Divisibility test
- A quick rule to check whether one number divides into another exactly.
- Multiple
- A number you get by multiplying, for example 3, 6, 9, 12 are multiples of 3.
- Factor
- A number that divides exactly into another number.
- Venn diagram
- A diagram of overlapping regions used to sort numbers by their properties.