Difference between revisions of "2013 AMC 8 Problems/Problem 15"

m (Solution)
m (Solution)
Line 17: Line 17:
  
 
<math>5^3 + 6^s = 1421\\
 
<math>5^3 + 6^s = 1421\\
125 + 6^s = 1296</math>
+
125 + 6^s = 1421\\
 +
6^s=1296</math>
  
 
To most people, it would not be immediately evident that <math>6^4 = 1296</math>, so we can multiply 6's until we get the desired number:
 
To most people, it would not be immediately evident that <math>6^4 = 1296</math>, so we can multiply 6's until we get the desired number:

Revision as of 14:53, 13 October 2014

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

$3^p + 3^4 = 90\\ 3^p + 81 = 90\\ 3^p = 9$

Therefore, $p = 2$.

$2^r + 44 = 76\\ 2^r = 32$

Therefore, $r = 5$.

$5^3 + 6^s = 1421\\ 125 + 6^s = 1421\\ 6^s=1296$

To most people, it would not be immediately evident that $6^4 = 1296$, so we can multiply 6's until we get the desired number:

$6\cdot6=36$

$6\cdot36=216$

$6\cdot216=1296=6^4$, so $s=4$.

Therefore the answer is $2\cdot5\cdot4=\boxed{\textbf{(B)}\ 40}$.

See Also

2013 AMC 8 (ProblemsAnswer KeyResources)
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. AMC logo.png