The number of terms common to two A.P.s 3, 7, 11, …,  407 and 2, 9, 16, …, 709 is

By inspection, first common term to both the series is 23. Second common term = 51.
Third = 79 and so on. These numbers form an A.P.  23, 51, 79, … 

Since ${\mathrm{T}}_{15}=23+14\left(28\right)=23+392=415>407$

 And ${\mathrm{T}}_{14}=23+13\left(28\right)=387<407$

 $\therefore$number of common terms = 14.