Question

Why does the sum of the digit if divisiblr by 3 is divisible by the whole number?

Answer 1

The key point which you seem to have noted is that 10≡1(mod3). This means that a number is congruent to the sum of its digits mod 3 because 10k≡1(mod3), which you also seem to have noted. Thus n is divisible by 3 (congruent to 0 mod 3) if and only if the sum of its digits is. The same is true for other remainders modulo 3; a number has a remainder of 1 when divided by 3 if and only if the sum of its digits does, etc.

Your observation that 10k≡1(mod3) is the reason why it has been commented that you have already proved the result.