Odd Man Out – Numbers of the form (2^n − 1): 3, 7, 15, 27, 63, 127, 255

Introduction / Context:
This odd-man-out test leverages the well-known pattern 2^n − 1: 3, 7, 15, 31, 63, 127, 255, … Only one term in the given list does not conform to this form, and that is the answer.

Given Data / Assumptions:

• Given sequence: 3, 7, 15, 27, 63, 127, 255
• Target form: 2^n − 1 for integer n ≥ 2

Concept / Approach:
Check each term against near powers of two. 3=4−1, 7=8−1, 15=16−1, 63=64−1, 127=128−1, 255=256−1. Only 27 is not one less than a power of two (the nearest is 32, giving 31).

Step-by-Step Solution:
3 = 2^2 − 1 ✔7 = 2^3 − 1 ✔15 = 2^4 − 1 ✔27 ≠ 2^n − 1 (since 2^5 − 1 = 31) ✖63 = 2^6 − 1 ✔; 127 = 2^7 − 1 ✔; 255 = 2^8 − 1 ✔

Verification / Alternative check:
Replacing 27 with 31 would produce a perfect run of 2^n − 1 terms.

Why Other Options Are Wrong:
7, 15, 63, and 127 all equal 2^n − 1 for some n; they are consistent with the rule.

Common Pitfalls:
Confusing 27 (which equals 3^3) with a 2-power-minus-one value.

Final Answer:
27