A number leaves remainder 3 when divided by 5. What remainder does the square of the same number leave upon division by 5?

Introduction / Context:
Remainders of powers are efficiently handled using modular arithmetic. Knowing the remainder of a number modulo 5 allows quick determination of the remainder of its square modulo 5 by squaring within the modular system.

Given Data / Assumptions:

• Let n ≡ 3 (mod 5), meaning n leaves remainder 3 upon division by 5.
• We seek n^2 mod 5.
• All operations are within integers modulo 5.

Concept / Approach:
Apply the rule: if n ≡ r (mod m), then n^2 ≡ r^2 (mod m). Therefore, it suffices to compute 3^2 modulo 5 and report the remainder. This avoids dealing with any specific value of n.

Step-by-Step Solution:

Verification / Alternative check:
Example: n = 8 (since 8 ≡ 3 mod 5). Then n^2 = 64, and 64 mod 5 = 4, confirming the result.

Why Other Options Are Wrong:
9 is the unsimplified square; modulo 5 it equals 4. 3 and 1 correspond to squaring 3 incorrectly or confusing with other residues; 2 is unrelated here.

Common Pitfalls:
Forgetting to reduce the squared value modulo 5, or mistakenly thinking the remainder remains 3 after squaring.

Final Answer:
4