Divisibility by 4

This is the second post in the Divisibility Rules Series. In the last post, we discussed about divisibility by 2. In this post, we discuss divisibility by 4.

Now, how do we know if a number is divisible by 4?

Four divides 100 because 4 \times 25 = 100. It is also clear that four divides 200, 300, 400 and all multiples of 100. Therefore, four divides multiples of 1000, 10 000, and 100 000. In general, 4 divides 10^n, where n is an integer greater than 1.

Now, how do we know if a number that is not a power of 10 is divisible by 4. Let us try a few examples.

Example 1: Is 148 divisible by 4? 148 is equal to 100 + 48 and 100 is divisible by 4. Since 48 is also divisible by 4, therefore, 148 id divisible by 4.

Example 2: Is 362 divisible by 4? 362 is equal to 300 + 62. Now, 300 is divisible by 4. Since 62 is not divisible by 4, therefore, 362 is not divisible by 4.

Example 3: Is 3426 divisible by 4? 3426 = 3400 + 26. Now, 3400 is divisible by 4 (it’s a multiple of 100), and 26 is not divisible by 4. Therefore, 3426 is not divisible by 4.

By now, you would have realized that we just test the last 2 digits of the numbers if we want to find out if it is divisible by 4: 148, 362, and 3426.

The Algebraic Explanation 

We can always expand any number as product of integers and powers of 10. For example 148 and 3426 maybe represented as follows:

In general, any whole number  a_{n}a_{n-1}\cdots a_2a_1a_0 with digits a_n, a_{n-1}, \cdots, a_2, a_1 and a_0 can be  expressed as

and all numbers at the left hand side of a_2(10^2) are divisible by 4 since they are multiples of 10^n where n is an integer greater than 1. Therefore, we just have to look at a_1(10^1) + a_0(10^0), or the last two digits (the tens and the ones) of the numbers to see if it is divisible by 4.

Hence, we have the following rule: A number is divisible by 4 if the last two digits are multiples of 4.

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4 thoughts on “Divisibility by 4

  1. Please check example no. 3
    Example 3: Is 1298 divisible by 4? ………. Now, ……….is divisible by (it’s a multiple of 100), and is not divisible by . Therefore, ………… is not divisible by…. .

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