AMC 8 2012 Problem 12

AMC8 2012 年第 12 题

余数与同余★★★☆☆

What is the units digit of

13^{2012}

的个位数是多少？

A. $\hspace{.05in}1$
B. $\hspace{.05in}3$
C. $\hspace{.05in}5$
D. $\hspace{.05in}7$
E. $\hspace{.05in}9$

正确答案 · Correct Answer

解析 · Solution

Solution 1

The problem wants us to find the units digit of

13^{2012}

, therefore, we can eliminate the tens digit of 13, because the tens digit will not affect the final result. So our new expression is

3^{2012}

. Now we need to look for a pattern in the units digit.

3^1 \implies 3

3^2 \implies 9

3^3 \implies 7

3^4 \implies 1

3^5 \implies 3

. We observe that the units digit recurs every four powers of three. Using this pattern, we subtract 1 from 2012 and divide by 4. The remainder is the power of three that we are looking for, plus one. 2011 divided by 4 leaves a remainder of 3, so the answer is the units digit of

3^{3+1}

, or

3^4

. Thus, the units digit of

13^{2012}

\textbf{(A)}1

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