Problem
NT-B1-M05-P007 The Equation \(5x+8y=67\)
#7
★★☆☆☆ Level 2 of 5
Find all positive integer solutions of \(5x+8y=67\).
Consider the equation modulo \(5\).
Modulo \(5\), \(8y\equiv67\), so \(3y\equiv2\pmod5\). Hence \(y\equiv4\pmod5\). Since \(8y<67\), we have \(y\le8\), so \(y=4\). Then \(x=\frac{67-32}{5}=7\). The answer is \((7,4)\).
Here the modulus leaves only one candidate.