Select the odd number from the following alternatives based on whether it is a prime or a composite number.

Introduction / Context:
This question checks your ability to recognize prime and composite numbers among larger three digit values. Prime recognition is a frequent topic in quantitative aptitude and number system questions, so quick and accurate identification is very useful.

Given Data / Assumptions:

• The four numbers provided are: 211, 287, 283, and 277.
• You must select the number that does not share the same prime or composite status as the others.
• Use the standard definition that a prime number has exactly two distinct positive factors.

Concept / Approach:
The plan is to classify each number as prime or composite. A composite number can be factored into smaller positive integers, whereas a prime number cannot be split in this way except as 1 and itself. By applying divisibility tests and checking possible factors up to the approximate square root, you can identify which numbers are prime and which are not.

Step-by-Step Solution:
Step 1: For 211, test divisibility by small primes like 2, 3, 5, 7, 11, and 13. It is not divisible by any of these, so 211 is a prime number. Step 2: Check 283. Again test divisibility by 2, 3, 5, 7, 11, and 13. None of these divide 283 exactly, so 283 is also a prime number. Step 3: Check 277 in the same way. It is not divisible by 2, 3, 5, 7, 11, or 13, so 277 is also a prime. Step 4: Now analyze 287. It is not even, so it is not divisible by 2. However, 287 divided by 7 equals 41, so 287 equals 7 * 41 and clearly has more than two factors. Thus, 287 is a composite number. Step 5: Since three numbers are prime and one is composite, the composite number 287 is the odd one out.

Verification / Alternative check:
You can quickly verify by using digit sum checks and basic divisibility rules. For example, 287 has a digit sum of 2 + 8 + 7 = 17, so it is not divisible by 3, but trial division by 7 reveals that 7 * 41 equals 287. In contrast, trial division of 211, 283, and 277 by primes up to their square roots does not yield any exact factors. This confirms that these three are primes and 287 is composite.

Why Other Options Are Wrong:

• 211: Since it has no divisor other than 1 and 211, it is prime and thus similar to 283 and 277.
• 283: This is another prime number and matches the majority pattern.
• 277: This is also a prime number and therefore not the odd choice.

Common Pitfalls:
Some learners avoid checking divisibility beyond a couple of small primes and may misclassify a number. Others might assume that a three digit number is probably composite without verification. Systematic checking, at least for primes like 2, 3, 5, 7, 11, and 13, is essential for accuracy in exam conditions.

Final Answer:
287