From the numbers 3, 19, 67 and 133, which is the odd one out based on primality and composite structure?

Introduction / Context:
This is a straightforward odd one out question that relies on basic number theory concepts, specifically prime and composite numbers. You are asked to identify which number among 3, 19, 67 and 133 does not share the same property as the others.

Given Data / Assumptions:
The numbers are:

• 3
• 19
• 67
• 133

We assume that three of these numbers are primes, while one is composite (that is, it has non trivial factors besides 1 and itself).

Concept / Approach:
A prime number has exactly two distinct positive divisors, 1 and the number itself. A composite number has additional divisors. To solve this question, we test each number to see whether it can be factored into smaller integers. The one that can be factored non trivially is the odd one out.

Step-by-Step Solution:
Step 1: Test 3. The only positive divisors of 3 are 1 and 3, so 3 is prime.Step 2: Test 19. This number has no divisors other than 1 and 19. It is not divisible by 2, 3, 5 or any integer up to the square root of 19, so it is prime.Step 3: Test 67. It is not divisible by 2, 3, 5 or 7. Since 7^2 = 49 and 11^2 = 121, we only need to test divisors up to 7. None of them divide 67, so 67 is also prime.Step 4: Test 133. We find that 133 = 7 * 19, so it has divisors 1, 7, 19 and 133, making it composite.

Verification / Alternative check:
We can further confirm 133's composite nature by simple divisibility tests. It is odd, so it is not divisible by 2. The sum of its digits is 1 + 3 + 3 = 7, not a multiple of 3, so it is not divisible by 3. However, 133 / 7 = 19, which is an integer. This quick test confirms that 133 has a factor 7 and therefore is not prime.

Why Other Options Are Wrong:
The numbers 3, 19 and 67 are all primes. They share the same fundamental property, while 133 does not. Choosing any of the primes as the odd one would leave a composite number grouped with three primes, which would not make sense given the usual structure of such questions.

Common Pitfalls:
Students sometimes confuse larger primes with composites because they are not familiar with prime tests beyond small numbers. It is helpful to remember that you only need to test for divisibility up to the square root of the number when checking for primality. Doing so quickly shows that 3, 19 and 67 have no small factors, while 133 does.

Final Answer:
The only composite number in the list, and therefore the odd one out, is 133.