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## Homework Statement

Prove by induction on n that [tex] 11^n - 4^n [/tex] is divisible by 7 for all natural numbers n (not including zero).

## Homework Equations

Principle of mathematical induction.

## The Attempt at a Solution

Of course I start by showing that the statement holds for n=1

[tex] 11^1 - 4^1 = 7 [/tex]

and

[tex] 7/7 = 1 \in \mathbb{Z} [/tex]

Then I go on to assume that the statement holds for some value k, which gives

[tex] \frac{11^k - 4^k}{7} = a \in \mathbb{Z} [/tex]

then I examine the situation for k+1

[tex] \frac{11^{k+1} - 4^{k+1}}{7} [/tex]

and I must show that this is also equal to an integer.

However I have tried rewriting this in many ways in order that I may be able to use the induction assumption. But I have had no luck. Any hints to help me from here?

Thanks.