Problem
NT-B2-M07-P012 Express One Variable
#12
★★★☆☆ Level 3 of 5
Find all positive integer pairs \((x,y)\) such that \(x^2=xy+x+y+11\).
Collect all terms with \(y\) on the right and look at when \(x+1\) divides the numerator.
The equation gives \(y(x+1)=x^2-x-11\). But \(x^2-x-11=(x+1)(x-2)-9\). Hence \(x+1\mid 9\). Since \(x>0\), \(x+1=3\) or \(9\). For \(x=2\), we get \(y=-3\), invalid. For \(x=8\), we get \(9y=45\), so \(y=5\). The answer is \((8,5)\).
This shows an alternative to factorisation: isolate one variable and obtain divisibility.