2012 AMC 8 Problem 12

Attempt Problem 12 of the 2012 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 8 solutions, or check the answer key.

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12.

What is the units digit of $13^{2012}?$

$1$

$3$

$5$

$7$

$9$

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We have to find $13^{2012} \mod 10.$ The following is true:

$\begin{align*}13^{2012} &\equiv 3^{2012} \mod 10\\&\equiv 81^{503} \mod 10\\&\equiv 1^{503} \mod 10\\&\equiv 1 \mod 10\end{align*}$

This means $13^{2012}$ has the same units digit as $1,$ so the units digit of $13^{2012}$ is $1.$

Thus, the answer is A .