Have you ever wondered why some numbers will divide evenly (without a remainder) into a number, while others will not? The Divisibility Rules help us to determine if a number will divide into another number without actually having to divide. There is a divisibility rule for every number. However, some of the rules are easier to use than others. For the rest, it might just be simpler to actually divide.

Let's look at three of the divisibility rules:

The Rule for 2 : Any whole number that ends in 0, 2, 4, 6, or 8 will be divisible by 2.

Example: 456,791,824

This is the number four hundred fifty-six thousand, seven hundred ninety-one, eight hundred twenty-four. We can tell if 2 divides into this number without a remainder by just looking at the last digit.

456,791,824 - the last digit is a 4. This means that the number is EVEN and 2 will divide into it without a remainder. 456,791,824 is divisible by 2.


Example: 34,807

Check out the last digit of this number. 34,807 - the last digit is a 7. This means that the number is ODD and 2 will not divide into it evenly. There will be a remainder. So 34,807 is not divisible by 2.

The Rule for 4 : If the last two digits of a whole number are divisible by 4, then the entire number is divisible by 4.


Example: 456,791,824

For this rule, we will look at the last two digits: 456,791,824. Does 4 divide evenly into 24? Yes. That means that 4 will also divide evenly into 456,791,824 and there will be no remainder.


Example: 723,810

Again, we will take a look at the last two digits: 723,810. Does 4 divide evenly into 10? No. That means that 4 will not divide evenly into 7223,810 and there will be a remainder.

The Rule for 8: If the last three digits of a whole number are divisible by 8, then the entire number is divisible by 8.


Example: 456,791,824

For this rule, we will look at the last three digits of the number: 456,791,824. Does 8 divide evenly into 824? YES, 8 goes into 824, 103 times without anything left over. So this number is divisible by 8.


Example: 923,780

Again, we will focus on the last three digits of the number: 923,780. Does 8 divide evenly into 780? NO, 8 goes into 780, 97 times with a remainder of 4. So this number is not divisible by 8

The rules for 2, 4, and 8 should all look similar. That is because these numbers are related. Think about the powers of 2.

2¹ = 2

2² = 4

2³ = 8

The exponent, or power of two, used is also the number of digits that we have to use when performing the test.

The Rule for 5: Number that are divisible by 5 must end in 5 or 0.


Example: 34,780

For this rule we just look at the last digit: 34,780. The last digit is a 0, so this number is divisible by 5.


Example: 13,569

Again, we will focus our attention on the last digit: 13,569. The last digit is a 9, so this number is not divisible by 5.

The Rule for 10: Numbers that are divisible by 10 need to be even and divisible by 5, because the prime factors of 10 are 5 and 2. Basically, this means that for a number to be divisible by 10, the last digit must be a 0.


Example:23,890

Take a look at the last digit: 23,890. The last digit is a 0. So this number is even and divisible by 5. That means that it is also divisible by 10.


Example: 85,395

Take a look at the last digit: 85,395. The last digit is a 5. So this number is odd and divisible by 5. Because the number is not even, it is not divisible by 10. It has to end in 0 to be divisible by 10.

Let's put it all together. Take a look at this example.

1,782,645,988

This number ends in 8....

It is divisible by 2, but it is not divisible by 5 or 10.

This number ends in 88....

It is divisible by 4 because 4 goes into 88 evenly.

This number ends in 988....

It is not divisible by 8 because 8 does not go into 988 evenly.

45,981,400

This number ends in 0....

It is divisible by 2, 5, and 10.

This number ends in 00....

It is divisible by 4. (You might be thinking, "Hey, 4 doesn't divide evenly into 00", but it shows that 4 needs to go into a multiple of 100. 4 goes into 100 evenly, so it will go into multiples of 100 evenly as well.)

This number ends in 400....

It is divisible by 8 because 8 will go into 400, 50 times with no remainder.

How are the divisibility rules helpful?

The divisibility rules help us to find factors of numbers. Instead of actually having to do the long division, we can use the rule to determine if the number is a factor first before dividing and getting an answer.