Divisibility Tests
Every question with a fully worked solution.
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Theory
A divisibility test tells you whether a number divides exactly, with no remainder, without doing the whole division. Numbers that divide exactly into a larger number are its factors (or divisors). This Year 7 maths page gives the tests for \(2\), \(3\), \(4\), \(5\), \(6\), \(8\), \(9\), \(10\) and \(11\), and how to combine them.
Divisible by means the division leaves remainder \(0\). \(2716\) is divisible by \(4\) because \(2716 \div 4 = 679\) exactly. We also say \(4\) is a factor of \(2716\).
Each test looks at one part of the number: the last digit for \(2\), \(5\) and \(10\); the last two digits for \(4\), or the last three digits for \(8\); the digit sum (add all the digits) for \(3\) and \(9\).
Why the last digits? \(2716 = 2700 + 16\). Every whole hundred divides by \(4\), so only the \(16\) matters. In the same way \(1000 = 8 \times 125\), so for \(8\) only the last three digits matter.
Combine two tests for a bigger number. Split it into two factors with no common factor except \(1\); the number must pass both tests: \(6 = 2 \times 3\), \(12 = 3 \times 4\), \(15 = 3 \times 5\), \(18 = 2 \times 9\), \(45 = 5 \times 9\).
Test for 11. Start with the first digit, then subtract the next, add the next, and so on. If the result is \(0\) or a multiple of \(11\), the number is divisible by \(11\): \(7392\): \(7 - 3 + 9 - 2 = 11\), so yes (\(7392 = 11 \times 672\)).
The tests:
| Divisible by | Test | Example |
|---|---|---|
| \(2\) | last digit even (\(0, 2, 4, 6, 8\)) | \(8392\): ends in \(2\) |
| \(3\) | digit sum divisible by \(3\) | \(4251\): \(4 + 2 + 5 + 1 = 12\) |
| \(4\) | last two digits divisible by \(4\) | \(7348\): \(48 = 4 \times 12\) |
| \(5\) | last digit \(0\) or \(5\) | \(6295\): ends in \(5\) |
| \(6\) | passes the tests for \(2\) AND \(3\) | \(2574\): even, digit sum \(18\) |
| \(8\) | last three digits divisible by \(8\) | \(5264\): \(264 = 8 \times 33\) |
| \(9\) | digit sum divisible by \(9\) | \(6453\): \(6 + 4 + 5 + 3 = 18\) |
| \(10\) | last digit \(0\) | \(9170\): ends in \(0\) |
How to test a number
- Choose the test (last digits, or digit sum).
- Apply it to the right part of the number.
- For a bigger divisor, split it into two factors with no common factor and apply both tests.
- Check by dividing if you like.
(i) \(2\)
(ii) \(3\)
(iii) \(4\)
(iv) \(6\)?
| \(3 + 4 + 8\) | \(=\) | \(15\) |
| \(348 \div 6\) | \(=\) | \(58\) |
(i) Yes: the last digit \(8\) is even (\(348 \div 2 = 174\)).
(ii) Yes: digit sum \(15\), and \(15 \div 3 = 5\) (\(348 \div 3 = 116\)).
(iii) Yes: \(48\) is divisible by \(4\) (\(348 \div 4 = 87\)).
(iv) Yes: it passes the tests for \(2\) and \(3\).
(ii) Is \(352\) divisible by \(8\)?
(iii) Is \(351\) divisible by \(9\)?
(iv) Is \(350\) divisible by \(10\)?
| \(352 \div 8\) | \(=\) | \(44\) |
| \(3 + 5 + 1\) | \(=\) | \(9\) |
(i) Yes: last digit \(5\) (\(345 \div 5 = 69\)).
(ii) Yes: the last three digits are \(352 = 8 \times 44\).
(iii) Yes: digit sum \(9\) (\(351 \div 9 = 39\)).
(iv) Yes: last digit \(0\) (\(350 \div 10 = 35\)).
(i) \(2\)
(ii) \(5\)
(iii) \(10\)?
(i) Even last digit: \(3750\) and \(2868\)
(ii) Last digit \(0\) or \(5\): \(3750\) and \(4915\)
(iii) Last digit \(0\): only \(3750\)
(i) \(3\)
(ii) \(9\)?
| \(7 + 4 + 2 + 2\) | \(=\) | \(15\) |
(i) Yes: \(15 = 3 \times 5\) (\(7422 \div 3 = 2474\)).
(ii) No: \(15\) is not a multiple of \(9\).
| \(5 + 7 + 4\) | \(=\) | \(16\) |
The last digit \(4\) is even, so the test for \(2\) always passes. For \(3\): \(16 + ? = 18\), \(21\) or \(24\), so the missing digit is \(2\), \(5\) or \(8\). Check: \(5724 \div 6 = 954\).
| \(3012 \div 12\) | \(=\) | \(251\) |
\(12 = 3 \times 4\). Multiples of \(4\): \(3004\) (digit sum \(7\)), \(3008\) (\(11\)), \(3012\) (\(6\), passes). The answer is \(3012\).
| \(2 + 3 + 8 + 5\) | \(=\) | \(18\) |
| \(7 - 3 + 9 - 2\) | \(=\) | \(11\) |
\(45 = 5 \times 9\): \(2385\) ends in \(5\) and its digit sum \(18\) is a multiple of \(9\), so yes (\(2385 = 45 \times 53\)). For \(11\): the result \(11\) is a multiple of \(11\), so yes.
Common pitfalls
Frequently asked questions
How do you know if a number is divisible by 3?
Add its digits. If the digit sum is divisible by 3, so is the number.
What is the test for 4?
The last two digits make a number divisible by 4.
What is the test for 6?
The number is even and its digit sum is divisible by 3.
What is the test for 9?
The digit sum is divisible by 9.
How do you test for 12?
Test for \(3\) and for \(4\).