AoPSWiki

Our Precalculus course starts on Dec. 4. Master trig, complex numbers, and vectors and matrices in 2 and 3 dimensions. Click here to enroll today!

Personal tools

Search

Toolbox

2002 AMC 10B Problems/Problem 14

From AoPSWiki

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 a lot of $0$ 's on the end. So, the sum of the digits of $N$ is $14\Longrightarrow\mathrm{ (B) \ }$

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2002_AMC_10B_Problems/Problem_14"

Our Precalculus course starts on Dec. 4. Master trig, complex numbers, and vectors and matrices in 2 and 3 dimensions. Click here to enroll today!

© Copyright 2008 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us