Question

N\(^2\) leaves a remainder of 1 when divided by 24. What are the possible remainders we can get if we divide N by 12?

• A. 1, 5, 7 and 11
• B. 1 and 5
• C. 5, 9 and 11
• D. 1 and 11

Hint:

For problems involving modulo arithmetic, remember to check possible values by testing small numbers that satisfy the given conditions. Squaring values can help reveal patterns for larger moduli.

Answer 1

The Correct Option is A

Given: \( N^2 \equiv 1 \pmod{24} \). This means that the square of \( N \) leaves a remainder of 1 when divided by 24. Now, we want to find the possible remainders when \( N \) is divided by 12. Step 1: Find the possible values of \( N \mod 24 \) The equation \( N^2 \equiv 1 \pmod{24} \) implies that \( N^2 - 1 \) is divisible by 24. Therefore, \( N^2 - 1 = (N - 1)(N + 1) \) is divisible by 24. This condition is satisfied when \( N \equiv 1, 5, 7, 11 \pmod{12} \). We check: - \( N \equiv 1 \pmod{12} \) implies \( N^2 \equiv 1^2 = 1 \pmod{24} \),
- \( N \equiv 5 \pmod{12} \) implies \( N^2 \equiv 5^2 = 25 \equiv 1 \pmod{24} \),
- \( N \equiv 7 \pmod{12} \) implies \( N^2 \equiv 7^2 = 49 \equiv 1 \pmod{24} \),
- \( N \equiv 11 \pmod{12} \) implies \( N^2 \equiv 11^2 = 121 \equiv 1 \pmod{24} \). Thus, the possible remainders when \( N \) is divided by 12 are 1, 5, 7, and 11. Final Answer: The correct answer is (a) 1, 5, 7 and 11.