Question

An ascending series of numbers satisfies the following conditions:

i. When divided by $3, 4, 5$ or $6$ , the numbers leave a remainder of $2$ . ii. When divided by $11$ , the numbers leave no remainder.

The $6^{\text{th}}$ number in this series will be:

Accepted Answer

C. 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 $60 	imes 11 = 660$. $N_n = 242 + 660(n-1)$. For $n = 6$: $N_6 = 242 + 660(5) = 242 + 3300 = 3542$.

Number Theory Remainders Series

Explanation

Practise this under exam conditions