A Number Is Divisible By 3 If The Sum Of The Digits Of The Number Is Divisible By 3

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 $10^k\equiv 1\pmod{3}$ is the reason why it has been. If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3.