Question 176
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 $|S(m, n)| \equiv r \pmod{d}$. Here, $|A|$ represents the number of elements in the finite set $A$.
$$
S(m, n) = \{(a, b) \in \mathbb{Z}^2 \mid 1 \le a \le m, 1 \le b \le n, \gcd(a, b) = 1\}.
$$
Prove: for any positive integers $d, r$, there exist integers $m, n$ not less than $d$, such that $|S(m, n)| \equiv r \pmod{d}$. Here, $|A|$ represents the number of elements in the finite set $A$.