Contest problems with powers

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Learn it Units digits of powers repeat in short cycles (2: 2, 4, 8, 6). To count digits, pair 2s with 5s to make 10s. To compare huge powers, rewrite them with a common exponent.

Example: Which is larger, $2^{100}$ or $3^{60}$?

Common exponent 20: $2^{100} = (2^5)^{20} = 32^{20}$.
$3^{60} = (3^3)^{20} = 27^{20}$.
$32 > 27$, so $2^{100}$ is larger.

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