Question

A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N > 2M$ and $M > 4.$ Each team plays a $76$ game schedule.

How many games does a team play within its own division?

$36$

$48$

$54$

$60$

$72$

Answer: B

Concepts:Diophantine Equation modular arithmetic

Difficulty rating: 1560

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Written solution:

Each team plays $3N$ games within its own division and $4M$ games against the other division, so $3N+4M=76.$

Since $N>2M$ , we have $3N>6M$ , and hence $76=3N+4M>10M$ . Thus $M<7.6$ . Together with $M>4$ , this gives $M\in\{5,6,7\}$ .

Reducing $3N+4M=76$ modulo $3$ gives $M\equiv1\pmod3$ , so $M=7$ . Therefore the team plays $4M=28$ non-division games and $76-28=48$ division games.

Thus, B is the correct answer.

2015 AMC 8 Problem 24

Answer: B

Difficulty rating: 1560

Video solution:

Written solution:

Problem 24 in Other Years