Divisibility Rules

Divisibility rules help you quickly figure out if one number can be divided evenly by another, without needing to perform long division.

These rules rely on certain number properties that make it easy to spot divisors with just a quick glance.

Let’s review the divisibility rules for some of the most common numbers: 2, 3, 4, 5, 9, 11, and 25.

Divisibility by 2

A number is divisible by 2 if its last digit is even. Even numbers end in 0, 2, 4, 6, or 8, ensuring divisibility by 2.

For example, 12 is divisible by 2 because it ends in 2.

Divisibility by 3 and 9

To check if a number is divisible by 3 or 9, add up all its digits. If the total is divisible by 3, then the whole number is as well; the same goes for 9.

For example, take the number 4455. The sum of its digits is $ 4 + 4 + 5 + 5 = 18 $. Since 18 is divisible by both 3 and 9, 4455 is also divisible by both.

Divisibility by 5

A number is divisible by 5 if it ends in 0 or 5. Any multiple of 5, when written in decimal, will end in one of these two digits.

For instance, 340 is divisible by 5 because it ends in 0.

Divisibility by 4 and 25

For divisibility by 4, look at the last two digits. If this two-digit number is divisible by 4, the whole number is as well.

For example, in 3124, the last two digits are 24, and since 24 is divisible by 4, so is 3124.

For 25, the same rule applies, but this time the last two digits should be divisible by 25.

In 175, the last two digits are 75, which is divisible by 25.

If a number ends in two zeros, it’s divisible by both 4 and 25, as with 5200.

Divisibility by 7

The rule for divisibility by 7 is a bit unique. Take the last digit, double it, and subtract that from the rest of the number. If the result is divisible by 7 (including 0), the original number is too.

If necessary, you can repeat this process until you reach a number that is clearly divisible by 7.

This rule works for numbers of any length and, if repeated, provides a quick way to verify divisibility by 7.

Divisibility by 8

To determine if a number is divisible by 8, start by doubling the third-to-last digit. Then, add this to the second-to-last digit and double the result again. Finally, add the last digit of the number. If the total is a multiple of 8, the original number is divisible by 8.

This method offers a clear, step-by-step way to check for divisibility by 8.

Divisibility by 11

The rule for 11 is a bit more involved; it requires calculating the difference between the sum of the digits in odd positions and the sum of the digits in even positions, counting positions from the right. If the difference is divisible by 11, then so is the number.

Consider the number 121. The sum of the digits in odd positions is 1 + 1 = 2, and the sum of the digit in the even position is 2. The difference is 0, which is divisible by 11, so 121 is divisible by 11.

This method applies to numbers of any length and can be repeated until you get a result that’s easy to evaluate.

Summary of Divisibility Rules

In summary, here are the main divisibility rules:

Divisible by	Condition	Example
2	Last digit is even	12
3 or 9	Sum of digits is divisible by 3 or 9	4455 4 + 4 + 5 + 5 = 18
4 or 25	Last two digits form a number divisible by 4 or 25	3124 24:4=6 1475 75:25=3
5	Last digit is 0 or 5	340 125
6	Divisible by both 2 and 3	84 84:2=41 84:3=28
7	Difference between the number without the units digit and twice the units digit is divisible by 7	329 32-(9×2)=14 14:7=2
8	Add the units digit, twice the tens digit, and four times the hundreds digit. If the resulting sum is divisible by 8, then so is the original number.	6432 2+(3×2)+(4×4)=24 24 ÷ 8 = 3
11	Difference between the sum of odd and even-position digits is divisible by 11	91828 9+8+8-1-2=22 22:11=2

These rules make calculations faster, letting you check for divisibility in large numbers without lengthy division.

They’re useful in various math problems and help save time in tackling complex calculations.