JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Sequences and Series

The three progressions consist of numbers $\equiv 3(\mathrm{mod}4)$, $\equiv 2(\mathrm{mod}3)$ and $\equiv 2(\mathrm{mod}5)$ respectively. The last two conditions combine to $\equiv 2(\mathrm{mod}15)$, and the smallest number of this form that is also $\equiv 3(\mathrm{mod}4)$ is $47$. Common terms then advance in steps of $\mathrm{lcm}(4,3,5)=60$, giving $47,107,167$; the next, $227$, exceeds the final term $197$ of the third progression. Hence the common terms are $47,107,167$, and their sum is $321$.