Common terms in two APs

First sequence: common difference $d_1 = 4$, last term $415$. Second sequence: common difference $d_2 = 5$, last term $464$.

The first common term is $19$. The common difference of common terms $= \text{LCM}(4, 5) = 20$.

Common terms form an AP: $19, 39, 59, \dots, T_k$. The maximum value $T_k \le \min(415, 464) = 415$. $19 + (k - 1)20 \le 415$ $20(k - 1) \le 396 \implies k - 1 \le 19.8 \implies k \le 20.8$

Thus, there are 20 common terms.