Divisibility Tests

In math, we often need to find out if large integers are divisible by certain smaller numbers. For example, when factoring a large integer to find its prime factorization using the division method, we need to start by dividing by smaller numbers like 2, 3, or other smaller numbers. We don't even necessarily have to start with only prime numbers but can use numbers such as 8, 9, or 10 just to make the integers we are dividing by more manageable.

This article discusses some tips on how to find if certain larger integers are divisible by the smaller numbers 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Divisibility by 2, 3, or 4

An integer is divisible by 2 if it is even. If it is even, its last digit will be 0, 2, 4, 6, or 8.

Example 1

3458 is divisible by 2 because it ends in the digit 8.

7385 is not divisible by 2 because it ends in the digit 5.

You can tell that an integer is divisible by 3 if the sum of its digits is divisible by 3.

Example 2

3483 is divisible by 3 because

$3+4+8+3=18$ and

$18÷3=6$ .

8249 is not divisible by 3 because

$8+2+4+9=23$

which is not divisible by 3.

Example 3

You can use this rule more than once in an extremely large number.

For example, to test the number 8,597,938, you can add the digits.

$8+5+9+7+9+3+8=49$

You can then add

$4+9=13$

Therefore, 8,597,938 is not divisible by 3 because 3 does not divide evenly into 13.

Example 4

Let's try that again with another very large number. Is 9,645,984 divisible by 3?

$9+6+4+5+9+8+4=45$

$4+5=9$

Since 9 is evenly divisible by 3, so is 9,654,984.

Because 4 goes evenly into 100, an integer is divisible by 4 if the last two digits are divisible by 4.

Example 5

Is 3,588 divisible by 4? The last two digits are 88, which is divisible by 4. Therefore, 3,588 is divisible by 4.

Is 8,427 divisible by 4? The last two digits are 27, which is not divisible by 4. Therefore, 8,427 is not divisible by 4.

One thing to note is that, while not every even number is divisible by 4, only even numbers are divisible by 4. No odd number is divisible by 4.

Divisibility by 5, 6, or 7

An integer is divisible by 5 if the last digit is either a 5 or a 0.

Example 6

Is 3,984,180 divisible by 5? Yes, because the last digit is 0.

Is 8,349 divisible by 5? No, because the last digit is not 0 or 5.

An integer that is divisible by both 2 and 3 is divisible by 6. So to determine if an integer is divisible by 6, you must run both the test for divisibility by 2 and the test for divisibility by 3 on it.

Example 7

Is 3,756 divisible by 6?

First, is it divisible by 2? Yes, because the last digit is 6, an even number.

Then, is it divisible by 3? Let's check. $3+7+5+6=21$ , which is divisible by 3. So the integer is divisible by 3.

Therefore, 3,756 is divisible by 6.

Example 8

Is 283,124 divisible by 6?

First, is divisible by 2? Yes, because the last digit is 4, an even number.

Then, it is divisible by 3? Let's check. $2+8+3+1+2+4=20$ , which is not divisible by 3. So the integer is not divisible by 3.

Therefore, 283,124 is not divisible by 6.

Unfortunately, there are no good divisibility tests for the number 7. The easiest way to find out if a large number is divisible by 7 is to use a calculator or to quickly do a long-division problem using the number 7.

Divisibility by 8, 9, or 10

Because 8 goes evenly into 1000, you can tell that an integer is divisible by 8 if the last three digits are divisible by 8.

Example 9

Is 835,136 divisible by 8? It is because $8\times 17=136$ .

Is 20,375 divisible by 8? You can use a shortcut here. Like the numbers 2, 4, and 6, no odd number is divisible by 8. So you can immediately see that since this number ends in a 5, it is not divisible by 8.

Is 20,374 divisible by 8? It is not, because 374 is not evenly divisible by 8.

Similar to the test for divisibility by 3, you can tell that an integer is divisible by 9 if the sum of the digits is divisible by 9.

Example 10

Is 379,481 divisible by 9? Let's check and see.

$3+7+9+4+8+1=32$

Since 32 is not divisible evenly by 9, 379,481 is not divisible by 9.

How about 848,376? Is this divisible by 9? Let's check and see.

$8+4+8+3+7+6=36$ . Since $\frac{36}{9}=4$ , so 848,376 is divisible by 9.

This is probably the easiest test of the bunch. If an integer ends in a 0, it is divisible by 10.

Example 11

Is 83,476,950 divisible by 10? Yes, it is, because it ends in a 0.

Is 4,569 divisible by 10? No, it is not, because it does not end in a 0.

Topics related to the Divisibility Tests

Mixed Numbers: Division

Dividing by a Fraction

Division Whole Numbers

Flashcards covering the Divisibility Tests

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Divisibility Tests

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

Get help learning about divisibility tests

Tutoring is an excellent way to help your student learn and remember the divisibility tests for the numbers 2 through 10. These shortcuts can save them a lot of time in their future years as a math student, but first, they must memorize the various rules of divisibility. A tutor can help your student memorize these by discerning their learning style and teaching them memory tips that make the best use of their learning style. A tutor will work with your student at their pace, taking extra time on the rules that they find challenging and skimming through the rules that they pick up quickly and easily. We'd be happy to set your student up with a qualified math tutor who can help them learn the divisibility tests and more math tips and tricks. To get started, contact Varsity Tutors today and speak with one of our helpful Educational Directors. We look forward to helping your student.