Problem
ALG-B3-M06-P004 Cyclic Recurrence
#4
★★☆☆☆ Level 2 of 5
Let \(f\) be defined on residues modulo \(7\) and suppose \(f(x+1)\equiv f(x)+1\pmod 7\). Prove that \(f(x)\equiv x+c\pmod 7\) for some \(c\).
Start with \(c=f(0)\).
Let \(c=f(0)\). By induction, \(f(1)\equiv c+1\), \(f(2)\equiv c+2\), and so on. Since all values are taken modulo \(7\), we get \(f(x)\equiv c+x\pmod 7\) for every residue \(x\).
Short task on finite domains and modular equalities.