Common Terms in Arithmetic Progressions

Let common terms satisfy $a_n = b_m$: $46 + 8n = 98 + 4m \implies 8n - 4m = 52 \implies 2n - m = 13 \implies m = 2n - 13$

Since $1 \le m \le 100$ and $1 \le n \le 100$: $1 \le 2n - 13 \le 100 \implies 14 \le 2n \le 113 \implies 7 \le n \le 56$

The common terms correspond to $n \in \{7, 8, \dots, 56\}$. Number of terms $N = 56 - 7 + 1 = 50$.

First common term = $a_7 = 46 + 8(7) = 102$. Last common term = $a_{56} = 46 + 8(56) = 494$.

$\text{Sum} = \frac{50}{2} (102 + 494) = 25 \times 596 = 14900$