How to find the mod of this large number

elementary-number-theory modular-arithmetic discrete-mathematics

The digit sum of $600\ldots 03$ is a multiple of $3$, so that monster is a multiple of $3$. When multiplying by $3$, the result is a multiple of $9$. However, the result is odd, so modulo $18$, the result is $9$ (not $0$).


Too long for the calculator, so use pencil and paper?

$$\quad\quad\quad6000006000600000600006006000000003$$ $$=3\times2000002000200000200002002000000001$$ $$=3\times(2\times1000001000100000100001001000000000+1)$$ $$\quad\quad=6\times1000001000100000100001001000000000+3,$$ so $$6000006000600000600006006000000003\times3$$ $$=(6\times1000001000100000100001001000000000+3)\times3$$ $$=18\times1000001000100000100001001000000000+9$$ $$\equiv9\pmod{18}.$$

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