Determining the general form of $10^x \bmod 210$ While solving a problem I came across solving $10^x\bmod 210$ for various values of $x$. It seems that the values repeat after an interval of 6 for $x\geq4$. Can any one explain how can solve this equation for some value of $x$ and show that it repeats at an interval of 6.
Note that $210 = 2\times3\times5\times7$ is a product of the first four primes.
Thanks

$10^x \equiv 0 \mod 2$ and $\mod 5$ for $x \ge 1$.
$10 \equiv 1 \mod 3$ so $10^x \equiv 1 \mod 3$ for all $x \ge 0$.
$10 \equiv 3 \mod 7$ and $3^6 \equiv 1 \mod 7$ so $10^{6k} \equiv 1 \mod 7$ for all $k$.  Moreover since $3^2$ and $3^3$ don't work, $10^x \equiv 1 \mod 7$ only for multiples of $6$.  If $x, y \ge 1$ and $x = y + 6 k$,
$10^x - 10^y = 10^y (10^{6k} - 1) \equiv 0 \mod 2, 5, 3$ and $7$, and therefore $\mod 210$. 
