2011 AMC 8 Problems/Problem 22

What is the tens digit of $7^{2011}$ ?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }7$

Solution

The first couple powers of $7$ are $7, 49, 343, 2401, 16807.$ As you can see, the last two digits cycle after every 4 powers. $7^{1}\ (\text{mod }100) \equiv 7^{5}\ (\text{mod }100) \equiv 7^{2009}\ (\text{mod }100).$ From there, we go two more powers. The last two digits are $43$ so the tens digit is $\boxed{\textbf{(D)}\ 4}$

2011 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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2011 AMC 8 Problems/Problem 22

Solution

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