Problem
NT-B2-M12-P006 When \(n\mid f(n)\)
#6
★★★☆☆ Level 3 of 5
Let \(f\in\mathbb Z[x]\). Prove that \(n\mid f(n)\) for all positive \(n\) if and only if \(f(0)=0\).
Use \(f(n)\equiv f(0)\pmod n\).
From \(n-0\mid f(n)-f(0)\), we get \(f(n)\equiv f(0)\pmod n\). If \(n\mid f(n)\) for all \(n\), then \(n\mid f(0)\) for all \(n\), hence \(f(0)=0\). Conversely, if \(f(0)=0\), then \(f(n)-f(0)\) is divisible by \(n\), so \(n\mid f(n)\).
One of the main templates of the module.