The idea, step by step
Some tests use the final digit
Divisible means a whole-number division leaves no remainder. A number is divisible by two when it is even: its final digit is 0, 2, 4, 6 or 8. A number is divisible by five if its final digit is zero or five. A number is divisible by ten if its final digit is zero.
These tests match familiar counting patterns. All multiples of ten end in zero because ten ones regroup into one ten. The other digits may be large, but the final digit settles these particular tests.
| Divisor | Final digit |
|---|---|
| 2 | 0, 2, 4, 6 or 8 |
| 5 | 0 or 5 |
| 10 | 0 |
Other tests use a digit sum
For divisibility by three, add the digits and test whether that sum is a multiple of three. In 246, the digit sum is 2 + 4 + 6 = 12, so 246 is divisible by three. For nine, the digit sum must be a multiple of nine.
To test six, check both two and three. A number such as 114 is even and has digit sum six, so it is divisible by six. Passing only one test is not enough. These shortcuts decide divisibility; they do not yet give the quotient.
A common trap
Using the final digit alone to test divisibility by three.
Instead: Use the sum of all digits. For 132, the final digit is two, but the digit sum is six, so the whole number is divisible by three.
See the thinking
Choose the final-digit tests
Which of 2, 5 and 10 divide 370 exactly?
- The last digit is zero, which is even, so two works.
- The last digit is zero, so five works.
- It is also zero, so ten works.
Test two different digit sums
Is 234 divisible by 3? Is it divisible by 9?
- Add the digits: 2 + 3 + 4 = 9.
- Nine is a multiple of three and of nine.
- Therefore both divisibility tests pass.
Check your understanding
Is 72 divisible by 6? Give both checks.
Show answer
Yes.
It is even, so divisible by two. Its digit sum is nine, so divisible by three. Both tests pass.
Your turn
6 questions · 10 marks · Work at your own pace.
Read the questions online
- Which of 2, 5 and 10 divide 145 exactly?1 mark
- Is 318 divisible by 3? Show the test.1 mark
- Is 427 divisible by 9? Show the test.1 mark
- Is 156 divisible by 6? Explain.2 marks
- Replace the square with one digit so 24□ is divisible by both 5 and 3. Find every possible digit.3 marks
- Noah says every number divisible by 3 is divisible by 6. Give a counterexample and explain.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Which of 2, 5 and 10 divide 145 exactly?1 mark
The final digit five passes the five test, but it is neither even nor zero.
5 only. - Is 318 divisible by 3? Show the test.1 mark
3 + 1 + 8 = 12, a multiple of three.
Yes. - Is 427 divisible by 9? Show the test.1 mark
4 + 2 + 7 = 13, not a multiple of nine.
No. - Is 156 divisible by 6? Explain.2 marks
The number is even.
1 + 5 + 6 = 12, so it is divisible by three as well.
Yes. - Replace the square with one digit so 24□ is divisible by both 5 and 3. Find every possible digit.3 marks
Divisibility by five allows zero or five.
240 has digit sum six, so it passes three. 245 has digit sum eleven, so it fails.
0. - Noah says every number divisible by 3 is divisible by 6. Give a counterexample and explain.2 marks
Nine is divisible by three but is odd, so it is not divisible by two or six.
For example, 9.
Curriculum reference & lesson scope
Australian Curriculum v9.0; SA Curriculum Mathematics 2026
Factor pairs; multiples; divisibility; factor/divisibility algorithms.
Official curriculum source ↗