Classification – Odd one out (primes ≡ 1 mod 4 vs ≡ 3 mod 4) Among these primes, exactly one leaves a remainder of 3 when divided by 4, while the others leave a remainder of 1. Identify the odd one out.

Introduction / Context:
Beyond simple prime vs composite, primes can be classified by residues modulo small integers. Here, the mod 4 classification separates one prime from the others.

Given Data / Assumptions:

• Options (all prime): 13, 17, 23, 37
• We compare each value modulo 4.

Concept / Approach:
Compute n mod 4. If n ≡ 1 (mod 4), it leaves remainder 1 when divided by 4; if n ≡ 3 (mod 4), remainder 3. Many number theory results partition primes this way.

Step-by-Step Solution:
13 ÷ 4 → remainder 1 → 13 ≡ 1 (mod 4).17 ÷ 4 → remainder 1 → 17 ≡ 1 (mod 4).37 ÷ 4 → remainder 1 → 37 ≡ 1 (mod 4).23 ÷ 4 → remainder 3 → 23 ≡ 3 (mod 4).

Verification / Alternative check:
Quick mental division confirms the residues unambiguously. Only 23 falls into the 4k + 3 class here.

Why Other Options Are Wrong:

• 13: 1 mod 4.
• 17: 1 mod 4.
• 37: 1 mod 4.
• None of these: There is exactly one 3 mod 4 prime (23).

Common Pitfalls:
Equating “odd” with “3 mod 4.” Many odd numbers are 1 mod 4; verify the exact remainder.

Final Answer:
23