How do you know a number is divisible by 9 if?

Answer 1

The easiest way is to test it. Divide the number by 9, and see if the result is an even number, or if there is a fractional amount in the answer. This is quite easy to do with a calculator, but can be done by hand as well, if you know how to do long division.

To test a number without doing the division:

Add the digits of the number together and if this sum is divisible by 9 then so is the original number.

The sum can be tested in the same way, so repeatedly add the digits of the sum until until a single digit remain. If this digit is 9 then the original number is divisible by 9; otherwise it gives the remainder when the original number is divided by 9.

This single digit result is called the digital root of the number.

A quick way of getting this single digit is that whilst summing the digits of the number if the sum ever exceeds 9 subtract 9 from the sum (which can be done by adding the two digits together); this means if the digit 9 appears in the number it can be ignored (unless the number is entirely made up of 9s, in which case the number is [clearly] divisible by 9).

For example is 413796453981 divisible by 9?

413796453981

4 + 1 = 5

5 + 3 = 8

8 + 7 = 15 → 1 + 5 = 6 (or 15 - 9 = 6)

next digit 9 can be ignored as: 6 + 9 = 15 → 1 + 5 = 6 (or 15 - 9 = 6)

6 + 6 = 12 → 1 + 2 = 3 (or 12 - 9 = 3)

3 + 4 = 7

7 + 5 = 12 → 1 + 2 = 3 (or 12 - 9 = 3)

3 + 3 = 6

next digit 9 can be ignored as: 6 + 9 = 15 → 1 + 5 = 6 (or 15 - 9 = 6)

6 + 8 = 14 → 1 + 4 = 5 (or 14 - 9 = 5)

5 + 1 = 6

→ 413796453981 is not divisible by 9 and has a remainder of 6 when the division is done.

Answer 2

A number is divisible by 9 of its digits add up to a multiple of 9.