Understand the idea.
Some tests use the final digit
A number is divisible by another when the division has no remainder. For , and , inspect the final digit: even digits pass the test, or passes the test, and only passes the test.
The final-digit tests work because all complete tens are already divisible by and . Only the leftover ones can prevent exact division. A large number ending in passes all three tests, whatever its earlier digits are.
| 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 or , add all the digits. If the sum is a multiple of , the original number is divisible by ; for , the sum must be a multiple of . These tests decide whether division is exact.
To test divisibility by , require both an even number and a digit sum divisible by . Passing just one is insufficient: is even but not divisible by , while passes the test but is odd.
See the method in action.
Choose the final-digit tests
Which of 2, 5 and 10 divide 370 exactly?
Worked solution
- 1
Test divisibility by two.
The final digit 0 is even, so division by 2 is exact.
- 2
Test divisibility by five.
The final digit is 0, one of the two permitted endings.
- 3
Test divisibility by ten.
A final 0 means there are no leftover ones after making tens.
2, 5 and 10 all divide 370 exactly.
Test two different digit sums
Is 234 divisible by 3? Is it divisible by 9?
Worked solution
- 1
Add every digit.
The tests for 3 and 9 use the digit sum, not just the last digit.
- 2
Test the sum for a factor of three.
The digit sum is a multiple of 3, so 234 is divisible by 3.
- 3
Test the sum for a factor of nine.
The digit sum is also a multiple of 9, so 234 passes both tests.
It is divisible by both 3 and 9.
Try it yourself.
Check your understanding
Is 72 divisible by 6? Give both checks.
Show answer & reasoning
Yes.
is even, and its digit sum is , a multiple of . Both conditions for divisibility by hold; indeed .
Keep practising
6 worksheet questions · 10 marks
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 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
, a multiple of three.
Yes. - Is 427 divisible by 9? Show the test.1 mark
, not a multiple of nine.
No. - Is 156 divisible by 6? Explain.2 marks
The number is even.
, so it is divisible by three as well.
Yes. - Replace the square with one digit so 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 ↗