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Divisibility Tests

Divisibility Tests

Is \(16,764\) divisible by \(4\)? There are several ways to go about answering this question. You could use a calculator and see if the answer is a whole number. You could also use long division, but that could take a long time. Is there any way to see the answer immediately?

Yes! That's what divisibility tests are for. This article will go through divisibility tests for \(2, 3, 4, 5, 6, 8, 9, 10\) and \(11\), and soon you'll be able to instantly recognise whether any number is divisible by these!

Divisibility Tests meaning

If a number is divisible by another number, that means the answer will be a whole number. A divisibility test tells you whether a number is divisible by another without actually dividing by that number.

If a number is divisible by another, then the dividend (the number we are dividing by) is also called a factor of the divisor (the number that you are dividing).

\(7\) is not divisible by \(2\) since \(7/2=3.5\).

\(52\) is divisible by \(2\) since \(52/2=26\).

What can you say about a number that is divisible by another number?

If a number is divisible by another number, the answer will be a whole number.

Divisibility test rules

There are some straightforward rules that you can easily remember to test for divisibility. Some of them are things you probably already know, like the fact that a number that is divisible by \(10\) ends in a \(0\). Others a bit more unusual. Keep reading to see the common divisibility tests.

Standard divisibility tests

The following are all the standard rules you need to know for divisibility tests for \(2, 3, 4, 5, 6, 8, 9, 10\) and \(11\).

Multiples of 2

You know if a number is a multiple of \(2\) if the final digit is also a multiple of \(2\), e.g. \(2,4,6,8\) or \(0\).

Is \(1,090,356\) a multiple of \(2\)?

Solution

Since the final digit is \(6\), which is a multiple of \(2\), then \(1000,356\) is a multiple of \(2\).

The test for divisibility by 2 is also the test for whether numbers are odd or even - if a number is divisible by 2, then it is even. If a number is not divisible by 2, then it is odd.

Multiples of 3

A number is divisible by \(3\) if the sum of the digits is a multiple of \(3\).

Is \(321,402\) exactly divisible by \(3\)?

Solution

The sum of the digits is \(3+2+1+4+0+2=12\). Since \(12\) is divisible by \(3\), so is \(321,402\).

Multiples of 4

This rule is slightly more difficult than multiples of \(2\), but follows a similar logic. You know a number is a multiple of \(4\) if the last two digits make up a multiple of \(4\).

Is \(14,534\) divisible by \(4\)?

Solution

The last two digits are \(34\), which is not a multiple of \(4\), therefore \(14,534\) is not divisible by \(4\).

Multiples of 5

Multiples of \(5\) are very straightforward: they end in either \(5\) or \(0\).

Is \(144\) divisible by \(5\)?

Solution

No, since \(144\) does not end in either \(5\) or \(0\).

Multiples of 6

There are two stages to this rule: use the divisibility test for \(2\), then the divisibility test for \(3\). If a number is divisible by both \(2\) and \(3\), then it is also divisible by \(6\).

Is \(47,420\) divisible by \(6\)?

Solution

\(47,420\) passes the divisibility test for \(2\) since it is an even number. The sum of the digits is \(4+7+4+2+0=17\). Since \(17\) is not divisible by \(3\), \(47,420\) is not a multiple of \(6\).

Figuring out whether a number is divisible by \(8\) is sort of like the divisibility check for \(6\), in that there is more than one thing that needs to be checked.

Multiples of 8

Divisibility tests for \(8\) follow the same logic as for \(2\) and \(4\), except the last \(3\) digits need to be a multiple of \(8\).

Is \(2,562\) a multiple of \(8\)?

Solution

\(8\) goes into \(560\) \(70\) times, therefore \(562/8\) is \(70\) remainder 2. Therefore, \(2,562\) is not a multiple of \(8\).

Multiples of 9

Just like the test for multiples of \(3\), a number is a multiple of \(9\) if the digits sum to a multiple of \(9\).

Is \(288\) divisible by \(9\)?

Solution

The sum of the digits is \(2+8+8=18\) which is divisible by \(9\), so \(288\) is also divisible by \(9\).

Multiples of 10

A number is a multiple of \(10\) if it ends in \(0\).

Multiples of 11

This test is a little more involved, but it's rather simple and clever! From right to left, you need to add the first digit, subtract the second digit, add the third... etc. and if the resulting number is divisible by \(11\), then the original number is also divisible by \(11\).

Is \(16,929\) divisible by 11?

Solution

To perform the test, we need to add and subtract alternate digits from right to left as follows: \(9-2+9-6+1=11\), therefore \(16,929\) is divisible by 11.

Another divisibility test example

More examples are always good!

Explain how you know whether \(492,132\) is divisible by \(6\).

Solution

You know if a number is divisible by \(6\) if it passes both divisibility tests for \(2\) and \(3\).

  • It passes the divisibility test for multiples of \(2\) since its last digit is an even number.
  • It passes the divisibility test for multiples of \(3\) since the sum of the digits, \(4+9+2+1+3+2=21\), is a multiple of \(3\).

Therefore \(492,132\) is divisible by \(6\).

Divisibility tests for prime numbers

There are quite a lot of tests to remember. An easy way to remember these tests is to learn the tests for prime numbers, each of which is unique. The remaining numbers which are not prime are actually an adaptation of the prime number tests.

Prime numberTestRelated tests for non-prime numbers
2The final digit is also a multiple of \(2\)\(4\): last two digits make up a multiple of \(4\)\(6\): divisible by both \(2\) and \(3\)
3The sum of the digits is a multiple of \(3\)\(6\): divisible by both \(2\) and \(3\)\(9\): The sum of the digits is a multiple of \(9\)
5The last digit is either \(5\) or \(0\)\(10\): last digit is \(0\)
11From right to left, add the first digit, subtract the second digit, add the third... etc. and the resulting number is divisible by \(11\)There are no related tests that you need to know.

If you enjoy modular arithmetic, it can also be used in divisibility tests.

Proving divisibility tests using modular arithmetic

It is possible to prove these tests using modular arithmetic. If you need to revise modular arithmetic, check out this article.

Let us demonstrate divisibility by \(3\).

Any number \(n\) with 5 digits \(abcde\) is divisible by \(3\) only if \(n \equiv 0\mod 3\).

Therefore, \(n=10,000a+1,000b+100c+10d+e=10^4a+10^3b+10^2c+10d+e\).

Keeping to \(\mod3\): \(10\equiv 1\mod 3\), therefore \(10^2\equiv 1^2=1\mod 3\)...

And so: \(n=(1a+1b+1c+1d+e)\mod 3\) \(=(a+b+c+d+e)\mod e\).

Therefore, \(a+b+c+d\equiv 0\mod 3\). In other words, \(a+b+c+d+e\) has to be a multiple of three.

This method can be summarised as the following definition:

A number with digits between 0 and 9, \(a_n, a_{n-1}...a_1,a_0\), can be written as follows: $$10^na_n+10^{n-1}a^{n-1}+...+10a_1+a_0$$

Let's summarize what you have learned so far.

Divisibility Tests - Key takeaways

  • Divisibility tests for...
    • \(2\): final digit is also a multiple of \(2\)
    • \(3\): sum of the digits is a multiple of \(3\)
    • \(4\): last two digits make up a multiple of \(4\)
    • \(5\): last digit is either \(5\) or \(0\)
    • \(6\): divisible by both \(2\) and \(3\)
    • \(9\): digits sum to a multiple of \(9\)
    • \(10\): last digit is \(0\)
    • \(11\): from right to left, add the first digit, subtract the second digit, add the third... etc. and the resulting number is divisible by \(11\)

Frequently Asked Questions about Divisibility Tests

A divisibility test tells you whether a number is divisible by another without actually having to perform a division.

There are rules for the numbers 2 to 11 for divisibility.

If a number is divisible by another, then the dividend (the number we are dividing by) is also called a factor of the divisor (the number that you are dividing).

A divisibility test tells you whether a number is divisible by another without actually dividing by that number.  

A number is divisible by 3 if the sum of the digits is a multiple of 3.

Final Divisibility Tests Quiz

Question

What is the divisibility test for 2?

Show answer

Answer

The final digit is also a multiple of \(2\)

Show question

Question

What is the divisibility test for 3?

Show answer

Answer

The sum of the digits is a multiple of \(3\)

Show question

Question

What is the divisibility test for 11?

Show answer

Answer

From right to left, add the first digit, subtract the second digit, add the third... etc. and the resulting number is divisible by \(11\)

Show question

Question

What is the divisibility test for 6?

Show answer

Answer

It is divisible by both \(2\) and \(3\)

Show question

Question

How do you know if \(n\) is divisible by \(m\)?

Show answer

Answer

\(\frac{n}{m}\) is a whole number

Show question

Question

The number \(246\) is...

Show answer

Answer

...divisible by both \(2\) and \(3\).

Show question

Question

What is the divisibility test for 9?

Show answer

Answer

The digits sum to a multiple of \(9\)

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Question

What is the divisibility test for 5?

Show answer

Answer

The last digit is either \(5\) or \(0\)

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Question

What is the divisibility test for 10?

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Answer

The last digit is \(0\)

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Question

The number 144 is...

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Answer

...divisible by 2, 3, 4 and 6.

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Question

What is the divisibility test for 4?

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Answer

The last two digits make up a multiple of \(4\)

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Question

The number 54 is...

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Answer

...divisible by 2, 3 and 9.

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Question

The number 12 is...

Show answer

Answer

...divisible by 6.

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Question

The number 6 is...

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Answer

Divisible by 2, 3 and 6.

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Question

The number 11 is...

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Answer

Prime.

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