Classification – Odd one out (perfect powers): Which number is unlike the others: 121, 144, 189, 169?

Introduction / Context:
Number classification commonly tests recognition of perfect powers (squares, cubes). A quick square check often isolates the outlier. Here, three items are perfect squares; one is not.

Given Data / Assumptions:

• Numbers: 121, 144, 189, 169.
• Check perfect-square status only; do not overcomplicate.

Concept / Approach:
Compute square roots where plausible and verify exactness. If three are n^2 and one is not, the non-square is the odd one out.

Step-by-Step Solution:
121 = 11^2 → perfect square.144 = 12^2 → perfect square.169 = 13^2 → perfect square.189 → 13^2 = 169 and 14^2 = 196, so 189 is between squares; it is not a perfect square.

Verification / Alternative check:
Prime factors: 121 = 11^2; 144 = 2^4 * 3^2; 169 = 13^2; 189 = 3^3 * 7 → not all prime exponents even, confirming it is not a square.

Why Other Options Are Wrong:

• 121, 144, 169 are all exact squares of consecutive integers (11, 12, 13), reinforcing their shared property.

Common Pitfalls:
Confusing near-squares with exact squares. Always check exactness, not proximity.

Final Answer:
189