MAT Exam 2022 Paper

Concept:If $N^2$ leaves remainder $1$ when divided by $24$, then $N$ must be odd and not divisible by $3$.Explanation:Given $N^2 \equiv 1 \pmod{24}$.So $24$ divides $N^2 - 1 = (N-1)(N+1)$.Since $24 = 2^3 \times 3$, first consider remainders modulo $12$.Because $N^2$ is odd, $N$ must be odd.Possible remainders when an odd number is divided by $12$ are: $1, 3, 5, 7, 9, 11$.Also, $N^2 \equiv 1 \pmod{3}$ means $N$ is not divisible by $3$.So exclude remainders $3$ and $9$.The remaining possible remainders are $1, 5, 7, 11$.Each of these works, for example $N = 5, 7, 11$ gives $25, 49, 121$, all leaving remainder $1$ when divided by $24$.Answer:Option A: $1, 5, 7$ and $11$.