In the number series 5, 4, 9, 32, 273, which number is the odd man out based on the property of being a prime versus a composite number?

Introduction / Context:
This question tests basic number theory ideas about prime and composite numbers. You are given a small set of numbers and asked to identify the odd man out. The quickest way to proceed is to classify each number as prime or composite and then see which one has a different classification from the rest.

Given Data / Assumptions:
The series of numbers is:

• 5
• 4
• 9
• 32
• 273

We assume that all numbers are positive integers and that the distinction between prime and composite numbers is the key idea being tested.

Concept / Approach:
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors, namely 1 and the number itself. A composite number has more than two positive divisors. To find the odd man out, we simply determine which of the numbers is prime and which are composite. If one number alone is prime and all others are composite, that prime will be the odd one out.

Step-by-Step Solution:
Step 1: Examine 5. Its only positive divisors are 1 and 5, so 5 is a prime number.Step 2: Examine 4. The divisors of 4 are 1, 2 and 4, so it has more than two divisors and is composite.Step 3: Examine 9. The divisors are 1, 3 and 9, again more than two, so 9 is composite.Step 4: Examine 32. The divisors include 1, 2, 4, 8, 16 and 32, so it is clearly composite.Step 5: Examine 273. It factors as 3 * 7 * 13, so it has many divisors and is also composite.

Verification / Alternative check:
To confirm, note that 5 is the only number in the list that cannot be broken down into a product of smaller natural numbers other than 1 and itself. Every other term has at least one non trivial factorisation, such as 4 = 2 * 2, 9 = 3 * 3, 32 = 2^5 and 273 = 3 * 7 * 13. This clearly separates 5 from the rest.

Why Other Options Are Wrong:
The numbers 4, 9, 32 and 273 are all composite and share the common feature of having more than two factors. Selecting any of them as the odd one out would leave the prime number 5 grouped together with composite numbers, which would not match the natural mathematical grouping of the set.

Common Pitfalls:
Some learners may attempt to spot more complicated patterns like powers or special factor relationships. While such patterns sometimes exist, in this question the simplest and cleanest distinction is between prime and composite numbers. Recognising prime numbers quickly is a useful exam skill.

Final Answer:
The only prime number in the series, and therefore the odd man out, is 5.