JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Sequences and Series

List the common terms of $3,7,11,\dots ,407$ (difference $4$) and $2,9,16,\dots ,709$ (difference $7$). The first value appearing in both is $23$, and thereafter common terms recur every $\mathrm{lcm}(4,7)=28$, giving $23,51,79,\dots$. These must not exceed $407$, the last term of the first progression, so from $23+28(k-1)\le 407$ we get $k-1\le 13.7$, hence $k=14$ common terms.