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

Using Carmitcheal function, $\lambda(21)=$lcm$(\lambda(3),\lambda(7))=$lcm$(2,6)=6$
and  $(10,21)=1,10^6\equiv1\pmod{21}$
$10^0\equiv1\pmod{210}$
As $a\equiv b\pmod m\implies a\cdot k\equiv b\cdot k\pmod{m\cdot k} $
$10^0\equiv1\pmod{21}\implies 10\equiv10\pmod{210}$
$10^1\equiv10\pmod{21}\implies 10^2\equiv100\pmod{210}$
$\displaystyle 10^2\equiv-5\implies 10^3\equiv-50\pmod{210}\equiv160$
$\displaystyle 10^3=10\cdot10^2\equiv10(-5)\equiv-8\implies 10^4\equiv-80\pmod{210}\equiv130$
$\displaystyle 10^4=(10^2)^2\equiv(-5)^2\equiv4\implies 10^5\equiv40\pmod{210}$
$\displaystyle 10^5=10\cdot10^4\equiv10\cdot4\equiv-2\implies 10^6\equiv-20\pmod{210}\equiv190$
$\displaystyle 10^6\equiv1\pmod{21}\implies 10^7\equiv10\pmod{210}$
So, for $10'$s power$(>0)$  the cycle is $\cdots,10,100,160,130,40,190,\cdots$
