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## Homework Statement

Let ##a, b, c \in \mathbb{N \setminus \{0 \}}##. Show that for all ##n \in \mathbb{Z}## we have

$$n^{11a + 21b + 31c} \equiv n^{a + b + c} \quad (mod \text{ } 11).$$

## Homework Equations

## The Attempt at a Solution

We have to show that ##11 | (n^{11a + 21b + 31c} - n^{a + b + c}) \iff \exists k \in \mathbb{Z} : n^{11a + 21b + 31c} - n^{a + b + c} = 11k.##

I tried showing that ##11 | n^{11a + 21b + 31c}## and ##11 | - n^{a + b + c}## and then conclude that ## 11 | (n^{11a + 21b + 31c} - n^{a + b + c})##. I also tried breaking down the variables in even and odd but that gave me too many cases which became tedious very fast. I also tried induction on a keeping b,c fixed.