$S_1 = \begin{Bmatrix} 3,7,11,15,19,23,\ldots,407 \end{Bmatrix}$
$S_2 = \begin{Bmatrix} 2,9,16,23,\ldots,709 \end{Bmatrix}$
$\text{common diff} (d_1) \text{ for } S_1 = 4$
$\text{common diff} (d_2) \text{ for } S_2 = 7$
Now, we try to make a new sequence which contains the common elements of the given two arithmetic progression $(AP)$ sequences.
$\text{first common term}(a) = 23$, this will be the first term in our new sequence which will also be an AP sequence and its common difference,
$d = LCM(d_1 , d_2) = LCM(4,7) = 28.$
$n^{th}$ term of our new sequence is given by,
$T_n = a + (n-1)*d$
$\quad = 23 + (n-1)*28$
$\quad = 23 + 28n - 28$
$\quad = 28n - 5$
For the last term of our sequence we get,
$28n - 5 = \min(407,709) = 407$
$\implies 28n = 407 + 5 = 412$
$n = \dfrac{412}{28}= 14.741$
$n$ must be integer and since we are finding number of present terms we must take $floor$ meaning the last term can be $407$ or some value lower than it. So, $n = 14.$
There are $14$ common terms.
Correct Answer: $B$