The number of terms common to the two A.P.’s $3,7,11,\dots ,407$ and $2,9,16,\dots ,709$ is ____________.

Common Difference of first series $=4$

Common difference of second series $=7$

First common term $=23$

Common difference $=L.C.M\left(7\times 4\right)=28$

To find number of common terms we put

Last term $\le 407$

$\Rightarrow 23+\left(n-1\right)\times 28\le 407$

$\Rightarrow \left(n-1\right)\times 28\le 384$

$\Rightarrow n\le 13.71+1$

$\Rightarrow n\le 14.71$

$\Rightarrow n=14$