Question 20
Let $A_1, A_2, \dots, A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$, where $0 \lt b \lt n$. Prove that
$$
b \le 2 \left( k - \frac{d_2}{\text{gcd}(d_1, d_2)} \right) - 1.
$$
$$
b \le 2 \left( k - \frac{d_2}{\text{gcd}(d_1, d_2)} \right) - 1.
$$