# Divisibility by 3 and 9

#### Divisibility by 9

All numbers written only with the digit 9 are divisible by 9
For example: 9, 99, 999, 99999

Lets us take any number , for example 324
324 can be written as a sum of hundreds, tens and ones like this:
324 = 300 + 20 + 4 or 324 = 100 + 100 + 100 + 10 + 10 + 4
But 100 = 99 + 1 and 10 = 9 + 1
Then 324 = 99 + 99 + 99 + 3 + 9 + 9 + 2 + 4 = (99 + 99 + 99 + 9 + 9)+ (3 + 2 + 4)
The sum in the first brackets is divisible by 9 because all the addends are divisible by 9. If the sum in the second brackets (3+2+4) is also divisible by 9 than the whole number 324 is divisible by 9.
Because the sum 3 + 2 + 4 is divisible by 9, 324 is also divisible by 9

But 3 + 2 + 4 represents sum of the numbers expressed with the digits of certain number and so:

Numbers divisible by 9 are only those with sum of their digits divisible by 9

For example 15948 is divisible by 9 because the sum of the digits (1 + 5 + 9 + 4 + 8) is divisible by 9 and 31409 is not divisible by 9 because the sum of the digits (3 + 1 + 4 + 0 + 9) is not divisible by 9

#### Divisibility by 3

9 is dividend by 3 =>

Every number divisible by 9 is divisible by 3

For example 7425 is divisible by 9. It is divisible by 3 too.

But not every number divisible by 3 is divisible by 9. For example 6, 12, 15, 21, 24, 30 are divisible by 3 but non of them is divisible by 9.

We proceed the same way to determine the divisibility by 3 like the divisibility by 9 and so:

Numbers divisible by 3 are only those with sum of their digits divisible by 3

For example:
58302 is divisible by 3 because the sum of the digits ( 5 + 8 + 3 + 0 + 2) is divisible by 3
69145 is not divisible by 3 because the sum of the digits ( 6 + 9 + 1 + 4 + 5) is not divisible by 3

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