Track / Stage 4 / 124 of 340 #384 of 1964

Problem 384

AMC 12 late, AIME early

Combinatorics Difficulty 4.8 Find the answer

6. In an $m$ row, $n$ column grid, there are $m n$ small squares with a side length of 1. Each small square is colored with one of three colors: red, yellow, or blue. It is known that each row of the grid has 6 red small squares, each column has 8 yellow small squares, and the entire grid has 15 blue small squares. If $n$ is a two-digit prime number, then, $m=$ $\ldots, n=$ $\qquad$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

$\text { 6. } m=17, n=13 \text {. }$

From the problem, we have $6 m+8 n+15=m n$ . Therefore,
$\begin{array}{l} (m-8)(n-6)=48+15=63 \\ =1 \times 63=3 \times 21=7 \times 9 . \end{array}$

Since $n$ is a two-digit prime number, we have $n=13, m=17$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.