A unified method for treating a linear congruence with constraints (Q2563988)

Let \(n\), \(r\), \(s\) be natural numbers and \[ x_1+\dots+x_s\equiv n\bmod r\tag{*} \] a linear congruence. A solution of (*) is an \(s\)-tuple of residue-classes \(\text{mod }r\). The author investigates solutions which satisfy some conditions on \(\text{gcd}(x_i,r)\). It turns out that the number of these solutions is given by some simple functions \(f(n,r)\) which are even arithmetical functions of the first variable and multiplicative functions of the other variable.