# Difference between revisions of "2015 AMC 8 Problems/Problem 24"

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\textbf{(E) } 72 | \textbf{(E) } 72 | ||

</math> | </math> | ||

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+ | ===Solution 1=== | ||

+ | Note that the equation rewrites to <math>3N+4M=76</math>. | ||

+ | |||

+ | Now remark that if <math>(m,n)</math> is a solution to this equation, then so is <math>(m+3,n-4)</math>. This is because <cmath>3(n-4)+4(m+3)=3n-12+4m+12=3n+4m=76.</cmath> Thus, we can now take an "edge case" solution and work upward until both conditions (<math>N>2M</math> and <math>M>4</math>) are met. | ||

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+ | We see by inspection that <math>(M,N)=(1,24)</math> is a solution. By the above work, we can easily deduce that <math>(4,20)</math> and <math>(7,16)</math> are solutions. The last one is the intended answer (the next solution fails <math>N>2M</math>) so our answer is <math>3N=\boxed{48\textbf{ (B)}}</math>. |

## Revision as of 17:00, 25 November 2015

A baseball league consists of two four-team divisions. Each team plays every other team in its division games. Each team plays every team in the other division games with and . Each team plays a 76 game schedule. How many games does a team play within its own division?

### Solution 1

Note that the equation rewrites to .

Now remark that if is a solution to this equation, then so is . This is because Thus, we can now take an "edge case" solution and work upward until both conditions ( and ) are met.

We see by inspection that is a solution. By the above work, we can easily deduce that and are solutions. The last one is the intended answer (the next solution fails ) so our answer is .