CN-TST 2023 Problem 4

Given $m,n\in\mathbb N_+,$ define $$S(m,n)=\left\{(a,b)\in\mathbb N_+^2\mid 1\leq a\leq m,1\leq b\leq n,\gcd (a,b)=1\right\}.$$ Prove that: for $\forall d,r\in\mathbb N_+,$ there exists $m,n\in\mathbb N_+,m,n\geq d$ and $\left|S(m,n)\right|\equiv r\pmod d.$