Divisibility by 4: How Do We Know If a Number Is Divisible by 4?
In this post, we are going to learn the criteria for divisibility of the number 4 and understand how they function.
The divisibility criteria for the number 4 are rules to know if a number can be divided by 4. They are simple to learn and their explanations are easy to understand.
How do we know if a number is divisible by 4?
If a number can be expressed by multiplying another number by 4, it is divisible by 4.
You need to know a couple multiplication properties: associative and distributive. If you do not understand them clearly, you can review them in this post.
Criteria for dividing one and two digit numbers by 4
First, we are going to learn how to determine if a one or two digit number meets the criteria for divisibility by 4. Easy: it is when we divide and see that the remainder is zero.
For example: Is 24 divisible by 4?
Yes, because when we divide 24 by 4, the quotient is 6 and the remainder is 0.
24 = 6 x 4
Criteria for dividing three and four digit numbers by 4
In order for a three or four digit number to be divisible by 4 it has to meet one of two conditions:
- The last two digits are zero.
- The last two digits are divisible by 4.
For example: Are 500 and 339 divisible by 4?
500 is divisible by four because its last two digits are zero.
339 is not divisible by four because 39 (its last two digits) is not.
Applying the rules we know to see if they are met or not, helps us to determine if a number is divisible by four. But we don’t know the reasoning, let’s continue and try to understand.
Explanation of the criteria for dividing a number by 4
How is it possible that two simple rules are able to tell us whether or not a number meets the criteria for divisibility by 4? Where do these rules come from?
The reason is very simple, and we are going to explain it in three steps.
- The number 100.
We start with the smallest number possible that has zero as the last two digits, 100. If we divide 100 by 4, the quotient is 25 and the remainder is 0. 100 is divisible by 4.
100 = 25 x 4
- Numbers larger than 100 with a zero in the last two digits.
All numbers that have zero in the last two digits can be expressed by multiplying another number by 100. We’ll pick one, for example, 4,300.
4,300 = 43 x 100
Since we know that 100 is divisible by four, we can also say that 4,300 is. Here is the mathematical explanation:
4,300 = 43 x 100 = 43 x (25 x 4) = (43 x 25) x 4 = 1,075 x 4
We can use the same operation for any number that has these characteristics. This way, we discover the first rule: any number that has zero as its last two digits is divisible by 4.
- Other numbers.
For all the other numbers, those that are larger than one hundred and those that do not have zero in the last two digits, we can apply a process similar to the one mentioned earlier. They can be expressed as the sum of a number with zero in the last two digits plus another number. Let’s take a random number, for example, 6,548.
6,548 = 6,500 + 48
Since we know that 6,500 is divisible by 4, we shouldn’t forget to see if 48 is as well. Well yes, the last two digits are divisible by 4.
48 = 12 x 4
So, we can express it in the following way:
6,548 = 6,500 + 48 = (65 x 100) + 48 =
= (65 x 25 x 4) + (12 x 4) = (1,625 x 4) + (12 x 4) =
= (1,625 + 12) x 4 = 1,637 x 4
This is how we understand the second rule: any number is divisible by 4 if its last two digits are divisible by 4.
We don’t need to go through all the steps in this process each time we need to know if a number is divisible by four. We have learned the necessary criteria for dividing a number by 4, but understanding it helps to know why the criteria exists, and if one day we forget any of them… I’m sure we will remember where they came from!