Whole Numbers

Divisibility Rules

A number is divisible by another number if it can be divided equally by that number; that is, if it yields a whole number when divided by that number. For example, 6 is divisible by 3 (we say "3 divides 6") because 6/3 = 2, and 2 is a whole number. 6 is not divisible by 4, because 6/4 = 1.5, and 1.5 is not a whole number.

It is often useful to know whether a number divides another number. To check divisibility, one can always do the division out by hand and see if the result is a whole number. However, if the number we are dividing is large, this becomes very difficult. There are some divisibility rules which make this task much easier--these rules allow us to determine whether a number is divisible by another number without having to carry out the division.

Divisibility by 1

A number does not change when it is divided by 1. Therefore, every whole number is divisible by 1.

Divisibility by 2, 4, and 8

All even numbers are divisible by 2. Therefore, a number is divisible by 2 if it has a 0, 2, 4, 6, or 8 in the ones place. For example, 54 and 2,870 are divisible by 2, but 2,221 is not divisible by 2.

A number is divisible by 4 if its last two digits are divisible by 4. For example, 780, 52, and 80,744 are divisible by 4, but 7,850 is not divisible by 4. To check whether a number is divisible by 4, just divide the last two digits of the number by 4. If the result is a whole number, then the original number is divisible by 4.

A number is divisible by 8 if its last three digits are divisible by 8. For example, 880 and 905,256 are divisible by 8 but 74,513 is not divisible by 8. To check divisibility by 8, divide the last three digits of the number by 8. If the result is a whole number, then the original number is divisible by 8.

Divisibility by 3, 6, and 9

A number is divisible by 3 if its digits add up to a number divisible by 3. For example, 671,451 is divisible by 3 because 6 + 7 + 1 + 4 + 5 + 1 = 24, and 24 is divisible by 3. 84,950 is not divisible by 3, because 8 + 4 + 9 + 5 + 0 = 26, and 26 is not divisible by 3.

A number is divisible by 6 if it is divisible by 2 and 3. In other words, a number is divisible by 6 if its digits add up to a number divisible by 3 and it ends with an even number. 671,451 (our previous example) is divisible by 3, but not by 2, because 1 is not an even number. Therefore, 671,451 is not divisible by 6. On the other hand, 789,312 is divisible by 6, because 7 + 8 + 9 + 3 + 1 + 2 = 30, which is divisible by 3, and its last digit is 2, which is even.

A number is divisible by 9 if its digits add up to a number divisible by 9. For example, 91,458 is divisible by 9 because 9 + 1 + 4 + 5 + 8 = 27 and 27 is divisible by 9. 18,453 is not divisible by 9 because 1 + 8 + 4 + 5 + 3 = 21 and 21 is not divisible by 9.

Divisibility by 5

A number is divisible by 5 if it ends in 0 or 5. For example, 875 and 600 are divisible by 5, but 76,508 is not divisible by 5.

Divisibility by 10

A number is divisible by 10 if it ends in 0. For example, 65,110 and 90 are divisible by 10, but 6,007 is not divisible by 10.

Note: Because a number divided by itself always yields 1, a number is always divisible by itself. For example, 7 is divisible by 7 and 8,374 is divisible by 8,374.

Note: No number is divisible by a number greater than itself.