Does 13 divide 6^13 + 33^49?
I can't seem to think of a way to make this number more manageable.
(2^13 * 3^13) + (3^49 * 11^49)
3^13 can be dropped off for our purposes.
2^13 + 3^36 * 11^49
...And then what. Hrm.
Does 13 divide 6^13 + 33^49?
I can't seem to think of a way to make this number more manageable.
(2^13 * 3^13) + (3^49 * 11^49)
3^13 can be dropped off for our purposes.
2^13 + 3^36 * 11^49
...And then what. Hrm.
$\displaystyle 13$ is prime and so $\displaystyle \phi(13)=12$. Also, $\displaystyle 33\equiv 7\text{ mod }13$ so that $\displaystyle 6^{13}+33^{49}\equiv 6^{13}+7^{49}\text{ mod }13$. Since $\displaystyle (13,7)=(13,6)=1$ we have that $\displaystyle 6^{13}+7^{49}=6\cdot 6^{\phi(13)}+7\cdot 7^{4\phi(13)}\equiv 6+7=13\equiv 0\text{ mod }13$. So, yes it is divisible.