Remainder of a square modulo 5 If a number leaves remainder 3 when divided by 5, what remainder will its square leave when divided by 5?

Introduction / Context:
Squaring a residue modulo a small base is a standard modular arithmetic task. By replacing the number with its remainder and then squaring under the same modulus, we get the new remainder efficiently.

Given Data / Assumptions:

• N ≡ 3 (mod 5).
• We require N^2 mod 5.
• Modular arithmetic properties apply: operations respect the modulus.

Concept / Approach:
If N ≡ r (mod m), then N^2 ≡ r^2 (mod m). Therefore compute 3^2 mod 5. No need to know N explicitly.

Step-by-Step Solution:
Given r = 3 under modulus 5.Compute r^2 = 3^2 = 9.Reduce modulo 5: 9 mod 5 = 4.Therefore, N^2 ≡ 4 (mod 5).

Verification / Alternative check:
Example: Let N = 8 (which is 3 mod 5). Then N^2 = 64; 64 mod 5 = 4. This confirms the general reasoning.

Why Other Options Are Wrong:
3 is the original residue, not the square's residue; 5 is not a valid remainder; 0 would require N to be a multiple of 5.

Common Pitfalls:
Forgetting to reduce after squaring; mixing up N mod 5 with N^2 mod 5; assuming residues remain unchanged after exponentiation.

Final Answer:
4