# Thread: proving an equation using congruences

1. ## proving an equation using congruences

prove that 11 divides 456^654 + 123^321.

I know that you can turn it into a congruence.

456^654 is congruent to -123^321 (mod 11)

But then I have no idea where to go.

2. $\displaystyle \phi(11)=10$
$\displaystyle 456^{10}\equiv 1\,mod\,11$
$\displaystyle 456^{650}\equiv 1\,mod\,11$
$\displaystyle 456\equiv 5\,mod\,11$
$\displaystyle 456^4\equiv5^4\equiv625\equiv9\,mod\,11$
$\displaystyle 456^{650}\cdot 456^4\equiv 1\cdot 9\equiv 9\,mod\,11$
same here
$\displaystyle 123^{320}\equiv 1\,mod\,11$
$\displaystyle 123^{320}\cdot 123 \equiv 1\cdot 123\equiv 2\,mod\,11$
so
$\displaystyle 456^{654}+123^{321}\equiv 2+9\equiv 0\,mod\,11$

### find the base so that given statement is true 245 456=654

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