1972 AHSME Problems/Problem 31

Problem

When the number $2^{1000}$ is divided by $13$ , the remainder in the division is

$\textbf{(A) }1\qquad \textbf{(B) }2\qquad \textbf{(C) }3\qquad \textbf{(D) }7\qquad \textbf{(E) }11$

Solution

By Fermat's little theorem, we know that $2^{100} \equiv 2^{1000 \pmod{12}}\pmod{13}$ . However, we find that $1000 \equiv 4 \pmod{12}$ , so $2^{1000} \equiv 2^4 = 16 \equiv 3 \pmod{13}$ , so the answer is $\boxed{\textbf{(C)}}$ .

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1972 AHSME Problems/Problem 31

Problem

Solution