MAT Exam 2023 Paper

Concept:The required number $N$ must be such that $N+7$ becomes a common multiple of $12$, $16$, $18$, and $21$.Explanation:First, find the LCM of $12$, $16$, $18$, and $21$.$12 = 2^2 \times 3$, $16 = 2^4$, $18 = 2 \times 3^2$, $21 = 3 \times 7$.So, $\text{LCM} = 2^4 \times 3^2 \times 7 = 1008$.Hence, $N+7$ must be divisible by $1008$.Check the given options by adding 7 to each number.$504+7=511$, not divisible by $1008$.$301+7=308$, not divisible by $1008$.$1513+7=1520$, not divisible by $1008$.$3017+7=3024=3\times1008$, which is divisible by $1008$.Therefore, $3017$ is the required number from the given options.Answer:Option C: $3017$.