2002 AMC 10B Problems/Problem 14

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Problem

The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is

$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35$

Solution

Taking the squareroot, $N=5^{64}\cdot 8^{25}=5^{64}\cdot 2^{75}=10^{64}\cdot 2^{11}$. This is $2048$ with 64 $0$'s on the end. So, the sum of the digits of $N$ is $14\Longrightarrow\mathrm{ (B) \ }$

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