Consider the series 15, 19, 27, 38, 55 and 75. Which number is the odd man out because it has a special power property that the others do not share?

Introduction / Context:
This problem gives a short list of numbers and asks you to find the odd man out. There is no obvious progression in the differences or ratios, so instead of looking for a series rule between consecutive terms, we should examine individual number properties such as being a perfect square, a perfect cube or having some special factorisation.

Given Data / Assumptions:
The numbers in the series are:

• 15
• 19
• 27
• 38
• 55
• 75

We assume that one of these numbers has a distinct mathematical property related to powers, while the others are simply composite or prime numbers without that special power structure.

Concept / Approach:
A good way to approach such questions is to test for perfect powers like squares and cubes. A perfect cube is a number of the form n^3 for some integer n. If only one number in the list is a perfect cube and all others are not, that number can be considered the odd man out based on this structural property.

Step-by-Step Solution:
Step 1: Check 15. Its prime factorisation is 3 * 5, so it is not a perfect square or cube.Step 2: Check 19. It is a prime number and cannot be expressed as n^2 or n^3 for an integer n greater than 1.Step 3: Check 27. We have 27 = 3 * 3 * 3 = 3^3, which is a perfect cube.Step 4: Check 38. It factors as 2 * 19 and has no exponent higher than 1, so it is not a perfect square or cube.Step 5: Check 55. It factors as 5 * 11, again with no repeated prime factor that would give a higher power.Step 6: Check 75. The factorisation is 3 * 5 * 5, so although it has a squared factor 5^2, the whole number is not a perfect square or cube because the exponent of 3 is only 1.

Verification / Alternative check:
We can summarise these observations: 27 is the only number that can be written exactly as n^3, with n = 3. All other numbers either have mixed prime factors with exponents 1 or have partial square factors but do not form a perfect square or cube. No other term in the list is a pure power like 2^3, 4^3 or 5^3, which confirms that 27 is structurally unique.

Why Other Options Are Wrong:
The numbers 38, 55 and 75 are ordinary composite numbers without being complete squares or cubes. They do not have that strong and unique power property. Removing any of them would still leave 27 as a perfect cube among the remaining terms, so they cannot be regarded as the odd ones out under this criterion.

Common Pitfalls:
Students sometimes focus only on differences between consecutive numbers, which can be messy in this case. When no clear pattern appears in differences or ratios, it is often better to examine properties of individual numbers such as primality, perfect squares or perfect cubes. That method quickly highlights 27 as the only perfect cube in the list.

Final Answer:
The odd man out, because it is the only perfect cube, is 27.