Problem
NT-B2-M01-P001 Constant Remainder
#1
★★☆☆☆ Level 2 of 5
Find all possible values of \( \gcd(n+4,n^2+2n+10) \), where \(n\) is a positive integer.
Replace \(n\) by \(-4\) in the second expression modulo \(n+4\).
Let \(d=\gcd(n+4,n^2+2n+10)\). Modulo \(n+4\), \(n\equiv-4\), so \(n^2+2n+10\equiv16-8+10=18\). Hence \(d=\gcd(n+4,18)\), so only the divisors of \(18\) are possible: \(1,2,3,6,9,18\). Each occurs: for a divisor \(r\mid18\), take \(n+4=r+18\). Then \( \gcd(n+4,18)=r \).
A good opening problem: students see that the gcd of a parametric expression can have a finite set of values.