Common Terms in Arithmetic Progressions

The first sequence has first term $17$ and common difference $4$. The second sequence has first term $16$ and common difference $5$.

The first common term is $21$.

The common difference of the common terms is $\text{LCM}(4, 5) = 20$.

The common terms form an AP: $21, 41, 61, \dots, T_k$.

The maximum possible term cannot exceed $\min(417, 466) = 417$.

$21 + (k - 1) \times 20 \le 417$ $20(k - 1) \le 396 \implies k - 1 \le 19.8 \implies k \le 20.8$

Therefore, there are $20$ common terms.