Number Theory Remainders Series

LCM$(3, 4, 5, 6) = 60$. Numbers leave remainder 2 when divided by 60, so $N = 60k + 2$. We need $60k + 2 \equiv 0 \pmod{11} \implies 5k + 2 \equiv 0 \pmod{11} \implies 5k \equiv 9 \pmod{11} \implies k \equiv 4 \pmod{11}$.

First value $k = 4 \implies N_1 = 60(4) + 2 = 242$. Subsequent values occur every $60 \times 11 = 660$. $N_n = 242 + 660(n-1)$. For $n = 6$: $N_6 = 242 + 660(5) = 242 + 3300 = 3542$.