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Powers of numbers behave in surprising ways...

Take a look at the following and try to explain what's going on.

This power pylon is unlikely to help...
Work out $2^1, 2^2, 2^3, 2^4, 2^5, 2^6$...
For which values of $n$ will $2^n$ be a multiple of $10?$

For which values of $n$ is $1^n + 2^n + 3^n$ even?

Work out $1^n + 2^n + 3^n + 4^n$ for some different values of $n$.
What do you notice?

What about $1^n + 2^n + 3^n + 4^n + 5^n?$

What other surprising results can you find?

Here are some suggestions to start you off:

$4^n + 5^n + 6^n$
$2^n+3^n$ for odd values of $n$
$3^n + 8^n$
$2^n + 4^n + 6^n$
$3^n + 5^n + 7^n$
$3^n  - 2^n$
$7^n + 5^n - 3^n$

Can you justify your findings?

You may also like to take a look at Big Powers.

Click here for a poster of this problem.