Problem
NT-B2-M07-P004 Divisibility by Reducing the Variable
#4
★★☆☆☆ Level 2 of 5
Find all positive integers \(n\) such that \(n+5\mid n^2+7\).
Work modulo \(n+5\): then \(n\equiv -5\).
Modulo \(n+5\), we have \(n^2+7\equiv 25+7=32\). Therefore \(n+5\mid 32\). Since \(n+5>5\), we get \(n+5=8,16,32\). Hence \(n=3,11,27\). Direct checking shows that all three values work.
Shows how a variable divisor becomes a fixed divisor.