Odd One Out — Consider the pairs (18, 45), (23, 14), (29, 82), (36, 27). Choose the pair that uniquely contains a perfect square and justify.

Introduction / Context:When several numerical relations seem plausible (gcd, parity mix, sum), a crisp discriminator is the presence of a perfect square. This item asks you to spot the only pair containing one.

Given Data / Assumptions:

• Pairs: (18, 45), (23, 14), (29, 82), (36, 27).
• Perfect square means n^2 for some integer n.

Concept / Approach:Scan each pair for 4, 9, 16, 25, 36, 49, 64, 81, … The first few squares quickly isolate candidates.

Step-by-Step Solution:18: not a square; 45: not a square.23: not; 14: not.29: not; 82: not.36: equals 6^2 (a perfect square); 27: not a square.Therefore, only (36, 27) contains a perfect square.

Verification / Alternative check:You might try gcd or “both multiples of 9” to compare pairs. Those checks split the set into two pairs each and do not yield a single outlier. The square test produces a unique selection.

Why Other Options Are Wrong:

• (18, 45): no perfect squares present.
• (23, 14): no perfect squares present.
• (29, 82): no perfect squares present.

Common Pitfalls:Overfitting with multi-criterion patterns. Favor single, decisive invariants that yield exactly one exception.

Final Answer:36, 27