2013 AMC 8 Problems/Problem 15

Problem

If $3^p + 3^4 = 90$ , $2^r + 44 = 76$ , and $5^3 + 6^s = 1421$ , what is the product of $p$ , $r$ , and $s$ ?

$\textbf{(A)}\ 27 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 90$

Solution

First, we're going to solve for $p$ . Start with $3^p+$ 3^4 $=90$ . Then, change $3^4$ to $81$ . Subtract $81$ from both sides to get $3^p=9$ and see that $p$ is $2$ . Now, solve for $r$ . Since $2^r+44=76$ , $2^r$ must equal $32$ , so $r=5$ . Now, solve for $s$ . $5^3+6^s=1421$ can be simplified to $125+6^s=1421$ which simplifies further to $6^s=1296$ . Therefore, $s=4$ . $prs$ equals $2*5*4$ which equals $40$ . So, the answer is $\boxed{\textbf{(B)}\ 40}$ .

2013 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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2013 AMC 8 Problems/Problem 15

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