Given a number with digits of the form

If the sum of these digits is equal to a number that is divisible by 3 then the number is divisible by 3. Proof follows

Starting with with which must be 0.3,6,9 then

is said to be divisible by 3 thus Where is an integer

should be divisible by 3.

Rearranging the terms of

Yields

Substituting in

Yields

Simplifying yields

And therefore it is a sufficient condition that if be divisible by 3 then is divisible by 3 and by extension is divisible by 3 if is divisible by 3.

As you might guess the same applies to all powers of 3 with 9 being the most useful next integer.

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