The number of common terms of the arithmetic progressions 3,7,11,…,407 and 2,9,16,…,709 is

A common term $N$ satisfies $N\equiv3\pmod4$ and $N\equiv2\pmod7$. Writing $N=4a+3$ and testing values with $N\equiv2\pmod 7$, the smallest is $N=23$ ($23=4\cdot5+3$ and $23=3\cdot7+2$). Because $\operatorname{lcm}(4,7)=28$, the common terms form the AP $23,51,79,\ldots$ with common difference $28$. The first progression stops at $407$, so we need $23+28(k-1)\le407$, i.e. $k-1\le\dfrac{384}{28}=13.71$, giving $k-1\le13$. The second progression extends to $709$, so $407$ is the binding limit. Hence there are $13+1=14$ common terms.