2023 Chinese IMO National Team Selection Test

For positive integers $m,n$, define $$S(m,n)=\{(a,b)\in {\mathbb{Z}}^2\mid 1\le a\le m,1\le b\le n,\gcd (a,b)=1\}.$$ Prove: for any positive integers $d,r$, there exist integers $m,n$ not less than $d$, such that $|S(m,n)|\equiv r(\mathrm{mod}d)$. Here, $|A|$ represents the number of elements in the finite set $A$.