Find the ordered pairs (m,n) where m, n are positive integers satisfying the following equation: 6mn = 27 + m - 2n.

Guest Apr 18, 2022

#1**+1 **

The best way to solve this problem is by using Simon's Favorite Factoring Trick.

We begin by isolating the coefficient. This gives: \(6mn+2n-m=27\)

Now, factor out \(n\): \(2n(3m+1)-m=27\)

We now need to factor out \(m\), in terms of \((3m+1)\), but, in order to do so, \(m\) must be a multiple of 3.

To achieve this, we multiply the entire equation by 3:\(6n(3m+1)-3m=81\)

Now, we can factor out m as so: \(6n(3m+1) -1(3m+1)=-1+81\)

Simplifying, we get: \((6n-1)(3m+1)=80\)

We now need integer factors, so we try out each of the factor pairs of 80.

The only successful integer solution is \(\color{brown}\boxed{(5, 1)}\)

BuilderBoi Apr 18, 2022