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 $10$.
b) $2^{1} + 3^{1}$, $2^{3} + 3^{3}$, $2^{5} + 3^{5}$, ......, $2^{99} + 3^{99}$ are all multiples of $5$.
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.
