Problem
NT-B1-M02-P019 Another Parameter GCD
#19
★★★☆☆ Level 3 of 5
Find all positive \(n\) such that \(\gcd(n^2+4,n+6)>1\).
Reduce \(n^2+4\) modulo \(n+6\).
Modulo \(n+6\), we have \(n\equiv -6\), so \(n^2+4\equiv36+4=40\). Thus the GCD is \(\gcd(n+6,40)\). It is greater than \(1\) if \(n+6\) is divisible by \(2\) or \(5\). Therefore \(n\) is even or \(n\equiv4\pmod5\).
Similar to the previous ones, but with a union of conditions.