Intermediate Math Question

Question

When is n² divisible by 3? Could you please find a pattern for n? After this, could you write a proof? Thanks!

Answer 1

3 divides n^2 when 3 divides n, when n = 0,3,6, 9, 12, 15, ....

then n^2 = 0,9,36,81, 144, 225, ...

3 divides any number whose digits sum to a number also divisible by 3

2000001 is divisible by 3, because the digits sum to 3, which is divisible by 3