Power Mad!

Powers of numbers behave in surprising ways.
Can you find convincing arguments that explain why all the statements below are true?

a) $2^{1}, 2^{2}, 2^{3},......, 2^{99}$ are never multiples of $10$.


c) $1^{99} + 2^{99} + 3^{99}$ is even

d) $1^{99} + 2^{99} + 3^{99} + 4^{99}$ is a multiple of $5$

e) $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is a multiple of $5$

f) $2^{99} + 3^{99} + 4^{99} + 5^{99} + 6^{99}$ is a multiple of $5$.

g) $3^{99} + 4^{99} + 5^{99} + 6^{99} + 7^{99}$ is a multiple of $5$.

h) $1^{x} + 2^{x} + 3^{x} + 4^{x} + 5^{x}$ is a multiple of $5$ when x is odd.


What other surprising results can you find? Can you explain why they are true?


Click here for a poster of this problem.