 Checking Divisibility by Adding Digits - dummies

Sometimes you can check divisibility by adding up all or some of the digits in a number. The sum of a number’s digits is called its digital root. Finding the digital root of a number is easy, and it’s handy to know.

To find the digital root of a number, just add up the digits and repeat this process until you get a one-digit number. Here are some examples:

The digital root of 24 is 6 because 2 + 4 = 6.

The digital root of 143 is 8 because 1 + 4 + 3 = 8.

The digital root of 51,111 is 9 because 5 + 1 + 1 + 1 + 1 = 9.

Sometimes you need to do this process more than once. Here’s how to find the digital root of the number 87,482. You have to repeat the process three times, but eventually you find that the digital root of 87,482 is 2:

8 + 7 + 4 + 8 + 2 = 29

2 + 9 = 11

1 + 1 = 2

Read on to find out how sums of digits can help you check for divisibility by 3, 9, or 11.

## Divisible by 3

Every number whose digital root is 3, 6, or 9 is divisible by 3.

First, find the digital root of a number by adding its digits until you get a single-digit number. Here are the digital roots of 18, 51, and 975:

18: 1 + 8 = 9

51: 5 + 1 = 6

975: 9 + 7 + 5 = 21; 2 + 1 = 3

With the numbers 18 and 51, adding the digits leads immediately to digital roots 9 and 6, respectively. With 975, when you add up the digits, you first get 21, so you then add up the digits in 21 to get the digital root 3.

Thus, these three numbers are all divisible by 3. If you do the actual division, you find that 18 ÷ 3 = 6, 51 ÷ 3 = 17, and 975 ÷ 3 = 325, so the method checks out.

However, when the digital root of a number is anything other than 3, 6, or 9, the number isnt divisible by 3:

1,037: 1 + 0 + 3 + 7 = 11; 1 + 1 = 2

Because the digital root of 1,037 is 2, 1,037 isnt divisible by 3. If you try to divide by 3, you end up with 345r2.

## Divisible by 9

Every number whose digital root is 9 is divisible by 9.

To test whether a number is divisible by 9, find its digital root by adding up its digits until you get a one-digit number. Here are some examples:

36: 3 + 6 = 9

243: 2 + 4 + 3 = 9

7,587: 7 + 5 + 8 + 7 = 27; 2 + 7 = 9

With the numbers 36 and 243, adding the digits leads immediately to digital roots of 9 in both cases. With 7,587, however, when you add up the digits, you get 27, so you then add up the digits in 27 to get the digital root 9. Thus, all three of these numbers are divisible by 9. You can verify this by doing the division: However, when the digital root of a number is anything other than 3, 6, or 9, the number isn’t divisible by 3. Here’s an example:

Because the digital root of 706 is 4, 706 isnt divisible by 9. If you try to divide 706 by 9, you get 78r4.