Divisibility by 9

How can we tell whether an integer is divisible by 9? It is similar to divisibility by 3: a number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits means the sum of all the digits of the number. Examples:

• Is 78,705 divisible by 9? The sum of its digits is 7 + 8 + 7 + 0 + 5 = 27. Since 27 is divisible by 9, 78,705 is divisible by 9 too.
• Is 11,288 divisible by 9? The sum of its digits is 1 + 1 + 2 + 8 + 8 = 20, and since 20 is not divisible by 9, 11,288 is not divisible by 9 either.

If necessary, we can take the sum of the digits more than once. For example, is 7,606,373,886 divisible by 9? The sum of its digits is 7 + 6 + 0 + 6 + 3 + 7 + 3 + 8 + 8 + 6 = 54. If we forgot whether 54 is divisible by 9, we could take the sum of the digits again: 5 + 4 = 9. This shows that 54 is divisible by 9, and so 7,606,373,886 is divisible by 9.