The Divisibility Tests: A Handy Guide for Numbers 1 to 10

When it comes to numbers, divisibility plays a crucial role in various mathematical operations. Whether you’re a student, a math enthusiast, or simply someone who wants to sharpen their mental math skills, understanding the divisibility rules can be immensely helpful. In this blog post, we will explore the divisibility tests for numbers from 1 to 10, providing you with a handy guide to make your calculations a breeze.

[Also Read: Understanding Integers]

Divisibility Tests for 2

Divisibility rules

Let’s start with the divisibility test for 2. Any number that ends in 0, 2, 4, 6, or 8 is divisible by 2. For example, 14 is divisible by 2 because it ends in 4, while 33 is not divisible by 2 because it ends in 3.

Example:
Let's consider the number 2468. 
Its last digit is 8, which is an even number. 
Therefore, according to the divisibility test for 2, the number 2468 is divisible by 2.

Similarly, let's consider the number 1357. 
Its last digit is 7, which is not even. 
Therefore, according to the divisibility test for 2, the number 1357 is not divisible by 2.

Divisibility Test for 3

Mental Math, Divisibility rules

Next, we have the divisibility test for 3. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3. For instance, let’s take the number 123. The sum of its digits is 1 + 2 + 3 = 6, which is divisible by 3, making 123 divisible by 3.

Example:
Consider the number 1236. 
The sum of its digits is 1 + 2 + 3 + 6 = 12. 
Since 12 is divisible by 3 (12 ÷ 3 = 4), the number 1236 is also divisible by 3.

On the other hand, let's take the number 457. 
The sum of its digits is 4 + 5 + 7 = 16. 
Since 16 is not divisible by 3, the number 457 is not divisible by 3.

Divisibility Test for 4

Mental Math, Divisibility rules

The divisibility test for 4 is based on the last two digits of a number. If the number formed by the last two digits is divisible by 4, then the entire number is divisible by 4. For example, the number 148 is divisible by 4 because 48 is divisible by 4.

Example:
Let's consider the number 1524. 
The last two digits are 24, and 24 is divisible by 4 (24 ÷ 4 = 6). 
Therefore, according to the divisibility test for 4, the number 1524 is divisible by 4.

Similarly, let's consider the number 3798. 
The last two digits are 98, and 98 is divisible by 4 (98 ÷ 4 = 24.5). 
Therefore, according to the divisibility test for 4, the number 3798 is divisible by 4.

Divisibility Test for 5

Mental Math, Divisibility rules

When it comes to testing divisibility by 5, all you need to remember is that any number ending in 0 or 5 is divisible by 5. For instance, 75 is divisible by 5 because it ends in 5, while 82 is not divisible by 5 because it ends in 2.

Example:
Consider the number 245. 
The last digit is 5, which satisfies the divisibility test for 5. 
Therefore, the number 245 is divisible by 5.

Similarly, let's take the number 1238. 
The last digit is 8, which does not satisfy the divisibility test for 5. 
Therefore, the number 1238 is not divisible by 5.

Divisibility Test for 6

Mental Math, Divisibility rules

The divisibility test for 6 combines the tests for divisibility by 2 and 3. If a number is divisible by both 2 and 3, then it is divisible by 6 as well. For example, the number 36 is divisible by both 2 and 3, making it divisible by 6.

Example:
Consider the number 252. 
It is divisible by both 2 (as the last digit is even) and 3 (since the sum of its digits is 2 + 5 + 2 = 9, which is divisible by 3). 
Therefore, the number 252 is divisible by 6.

Similarly, let's take the number 378. 
It is divisible by both 2 (as the last digit is even) and 3 (since the sum of its digits is 3 + 7 + 8 = 18, which is divisible by 3). 
Therefore, the number 378 is divisible by 6.

Divisibility Test for 7

Mental Math, Divisibility rules

The divisibility test for 7 is a bit trickier. To check if a number is divisible by 7, double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then the original number is divisible by 7. Let’s take the number 154 as an example. Double the last digit (4 x 2 = 8) and subtract it from the remaining number (15 – 8 = 7), which is divisible by 7. Hence, 154 is divisible by 7.

Example:
Consider the number 532.

The last digit is 2. Double it to get 4.
Remove the last digit from 532, which gives us 53.
Subtract 4 from 53, which results in 49.
Since 49 is divisible by 7 (7 * 7 = 49), the original number 532 is divisible by 7.

Similarly, for a number like 259:

The last digit is 9. Double it to get 18.
Remove the last digit from 259, which gives us 25.
Subtract 18 from 25, which results in 7.
Since 7 is divisible by 7 (1 * 7 = 7), the original number 259 is divisible by 7.

This method may seem a bit intricate compared to other divisibility tests, but it provides a straightforward way to determine divisibility by 7 without performing long division.

Divisibility Test for 8

Mental Math, Divisibility rules

Similar to the divisibility test for 4, the divisibility test for 8 focuses on the last three digits of a number. If the number formed by the last three digits is divisible by 8, then the entire number is divisible by 8. For example, the number 1,232 is divisible by 8 because 232 is divisible by 8.

Example:
Consider the number 1216. 
The last three digits are 216, and 216 is divisible by 8 (216 ÷ 8 = 27). 
Therefore, according to the divisibility test for 8, the number 1216 is divisible by 8.

Similarly, let's take the number 43568. 
The last three digits are 568, and 568 is divisible by 8 (568 ÷ 8 = 71). 
Therefore, according to the divisibility test for 8, the number 43568 is divisible by 8.

Divisibility Test for 9

Mental Math, Divisibility rules

For the divisibility test of 9, we follow a similar rule as the divisibility test for 3. If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. For example, let’s consider the number 567. The sum of its digits is 5 + 6 + 7 = 18, which is divisible by 9, making 567 divisible by 9.

Example:
Consider the number 891. 
The sum of its digits is 8 + 9 + 1 = 18. 
Since 18 is divisible by 9 (18 ÷ 9 = 2), the number 891 is also divisible by 9.

Similarly, let's take the number 2349. 
The sum of its digits is 2 + 3 + 4 + 9 = 18. 
Since 18 is divisible by 9 (18 ÷ 9 = 2), the number 2349 is also divisible by 9.

Divisibility Test for 10

Mental Math, Divisibility rules

Finally, the divisibility test for 10 is quite straightforward. Any number that ends in 0 is divisible by 10. For example, 150 is divisible by 10 because it ends in 0, while 127 is not divisible by 10 because it does not end in 0.

By mastering these divisibility tests for numbers 1 to 10, you can quickly determine whether a number is divisible by another without the need for long division or calculators. These tests are not only valuable in mathematical calculations but also in problem-solving and critical thinking. So, the next time you encounter a number, put these divisibility tests to use and impress yourself with your mental math skills!

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