Why does $\displaystyle 2^{10}\equiv1mod(11)$ and $\displaystyle 2^{30}\equiv1mod(31)$ imply that

$\displaystyle 2^{341}-2$ is divisible by both 11 and 31?

Thanks for any help

Is it because:

$\displaystyle 2^{341}={2^{10}}^{30}*2^{41}\equiv2^{41}={(2^{10}) }^4*2\equiv2mod(341)$