Divisibility Rules

What are Divisibility Rules?

Divisibility rules give you an easy method to determine whether a given integer (whole number) is divisible by some other specified smaller integer (generally a number between 2 and 12). The term “is divisible by” means that when we divide the first number (the larger) by the second number (the smaller), the remainder is zero. For example, 18 is divisible by 2, but 19 is not divisible by 2. Generally, divisibility using these rules can be determined in your head without the aid of pencil and paper.

Here are the rules for determining divisibility by the numbers 2 through 12. Most of these rules are easy to remember and can help you answer certain math questions quicker and more easily.

Divisibility by 2:
If the last digit is even (this is ends in 0, 2, 4, 6, or 8), then the number is divisible by 2. Otherwise it is not divisible by 2. For example, 23678 is divisible by 2, but 23677 is not divisible by 2.

Divisibility by 3:
If the sum of the digits is divisible by 3, then the number itself is divisible by 3. For example the number 321 is divisible by 3 since the sum of the individual digits is 6, and 6 is divisible by 3.

Divisibility by 4:
If the last two digits form a number that is divisible by 4, then number itself is divisible by 4. For example, the number 1236 is divisible by 4, since the number 36 is divisible by 4.

Divisibility by 5:
If the last digit of the number is a 0 or a 5, then the number is divisible by 5. For example, both the number 165 and 14550 are divisible by 5. However, 55003 is not divisible by 5.

Divisibility by 6:
If the number is divisible by both 3 and 2, it is also divisible by 6. For example, the number 144 is divisible by 2 (144 is an even number) and is divisible by 3 since the sum of the digits is 9 and 9 is divisible by 3. Therefore the number 144 must be divisible by 6.

Divisibility by 7:
To determine if a number is divisible by 7, take the last digit of the number, double it, and subtract it from the rest of the number. If the resulting number is divisible by 7, then the original number must also be divisible by 7. You may want to apply this rule multiple times to determine whether a specific number is divisible by 7. This rule is not as simple as the others, so let’s look at an example. Suppose we want to determine whether the number 2163 is divisible by 7. Actually, it is easy to see that this number is divisible by 7 but let’s apply the rule and see what we get. We take the last digit which is a 3 and double it to get 6. We then subtract 6 from the number formed by dropping that last digit, so we have 216 – 6 = 210. While it is fairly clear that 210 is divisible by 7, we could take this one step further and apply the rule again. We take the last digit of the number 210 which is 0, double it to get 0 and then perform the subtraction 21 – 0 = 21, which we know to be divisible by 7. All of this implies that our original number 2163 must be divisible by 7.

Divisibility by 8:
If the last three digits of the number form a number that is divisible by 8, then the original number must also be divisible by 8. For example, 82168 must be divisible by 8 since the number 168 is divisible by 8.

Divisibility by 9:
If the sum of the digits is divisible by 9, then the number itself is divisible by 9. For example, the number 459 must be divisible by 9 since the sum of the digits is 18, which is divisible by 9.

Divisibility by 10:
If the number ends in 0, it is divisible by 10. For example, the number 1234567890 must be divisible by 10.

Divisibility by 11:
To determine if a number is divisible by 11, alternately subtract and add the digits that make up the number from left to right. If the result is divisible by 11, the number is also (note that zero is divisible by all numbers). For example, applying this rule to the number 121 (which is 11 times 11), we have 1 – 2 + 1 = 0. Since 0 is divisible by 11, we conclude that 121 must be divisible by 11 (which we already knew). So let’s try a tougher one. We want to know if the number 365167484 is divisible by 11. We have 3 – 6 + 5 – 1 + 6 – 7 + 4 – 8 + 4 = 0. Therefore, the number 365167484 is divisible by 11.

Divisibility by 12:
If the number is divisible by both 3 and 4, it must also be divisible by 12. For example, the number 144 is divisible 3 since the digits sum to 9 and we know that 9 is divisible by 3. But 144 is also divisible by 4 since 4 divides 44 (the last two digits of the number 144). So we conclude that 144 must be divisible by 12.

How Are Divisibility Rules Useful?

There are several ways that knowledge of these divisibility rules can be useful. One of these is in determining whether a given number is a prime number (that is to say that the only integers that divide the given number are 1 and the number itself). Let’s look at an example. Suppose that you were asked to determine whether the number 107 is a prime number. You could break out a pencil and paper and start doing divisions or you could simply use the divisibility rules. To determine whether this number is a prime number we need to know whether and prime number less than 11 divides 107. We know that the number 107 is not divisible by two since it does not end in 0, 2, 4, 6, or 8. We know that the number 107 is not divisible by 3, since 1 + 0 + 7 totals 8, which is not divisible by 3. Next, we know that it is not divisible by 5 since the number 107 does not end in a 0 or a 5. So we move on to 7, where we take the last digit, a 7, double it to get 14, subtract 14 from 10 to get negative 4, which is not divisible by 7, so we conclude that 107 must not be divisible by 7. Since 7 is the last prime number less than 11, we can conclude that the number 107 must be a prime number.

Another place where divisibility rule can useful is when you want to find common factors in two or more numbers. For example, if we need to reduce a fraction that has a relatively large numerator and denominator. Suppose you were asked to reduce the fraction 129/870, how would you approach the problem? First, we notice from the divisibility rules we learned that 2 is not a common factor of 129 and 870 because it is not a factor of 129. However, we can quickly see from our rules that 3 is a common factor, so the problem reduces to 43/290.

Practice, Practice, Practice

The only way to get good at applying the divisibility rules is to practice using these rules. We provide a few practice exercises here, but you can easily make up your own practice exercises. Apply the divisibility rules to answer the following questions.

Exercise 1:

Is the number 1365 divisible by 2? Why or why not?

Exercise 2:

Is the number 1365 divisible by 3? Why or why not?

Exercise 3:

Is the number 1366 divisible by 4? Why or why not?

Exercise 4:

Is the number 1365 divisible by 5? Why or why not?

Exercise 5:

Is the number 1366 divisible by 6? Why or why not?

Exercise 6:

Is the number 1366 divisible by 7? Why or why not?