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If x and y are positive integers for which 3x + 2y + xy = 115 + 14x + 4y - 2x + 5y + 3, then what is x + y?

 Apr 26, 2022
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Simplify to: \(-9x-7y+xy=118\)

 

Factor out the y: \(y(-7+x)-9x=118\)

 

Add 63 to both sides so we can apply Simon's favorite factoring trick: \(y(-7+x)-9x+63=118\)

 

Furthur factor: \((y-9)(-7+x)=181\)

 

181 is prime, so the only factors are 1 and 181

 

There are 2 cases for x and y for which the equation is satisfied: \((8,190)\) and \((10, 188)\)

 

Coincidentally, both sum to \(\color{brown}\boxed{198}\)

 Apr 26, 2022

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