Problem
ALG-B3-M06-P013 Additivity Modulo a Prime
#13
★★★★☆ Level 4 of 5
Let \(p\) be prime, \(f:\mathbb Z/p\mathbb Z\to\mathbb Z/p\mathbb Z\), and \(f(x+y)=f(x)+f(y)\). Prove that \(f(x)=ax\) for some residue \(a\).
Every residue is a sum of several ones.
Let \(a=f(1)\). For a residue \(x\), we have \(x=1+\cdots+1\) modulo \(p\). Then \(f(x)=xf(1)=ax\). This also holds for \(x=0\), since \(f(0)=0\).
Basic fact for later modular problems.