2013 AMC 8 Problems/Problem 15

Problem $6\cdot6=36$

Solve for s, x, and r.

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

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

Problem

Solve for $p$ , $r$ , and $s$ . 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
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2013 AMC 8 Problems/Problem 15

Problem

Solution

See Also