Divisibility by 3 and 9

Divisibility by 9

All numbers that contain only the digit 9 are divisible by 9.
For example: 9, 99, 999, 99999

Lets us take a number, for example, 324
324 can be written as a sum of hundreds, tens and ones:
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 inside 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, then the whole sum, 324, is divisible by 9.
Since the sum 3 + 2 + 4 is divisible by 9, we conclude that 324 is also divisible by 9.

However, 3 + 2 + 4 is the sum of the digits in our number, hence the rule:

A number is divisible by 9 if the sum of its digits is divisible by 9.

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

Divisibility by 3

9 is divisible by 3 =>

Every number divisible by 9 is divisible by 3.

For example, 7425 is divisible by 9, hence it is divisible by 3.

However, a number divisible by 3 is not necessarily divisible by 9. For example 6, 12, 15, 21, 24, 30 are all divisible by 3 but none of them is divisible by 9.

The rule for divisibility by 3 can be easily obtained following the same logic we used with divisibility by 9.

A number is divisible by 3 if the sum of its digits is divisible by 3.

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


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