Which one of the following numbers is not a prime number?

Introduction / Context:
Prime numbers are fundamental in number theory and aptitude exams. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors, namely 1 and the number itself. This question asks you to identify which of the given numbers fails this property and therefore is not prime. Being able to quickly factor or test divisibility is very useful in competitive exams.

Given Data / Assumptions:

• We are given the numbers 91, 71, 41 and 31 (and an extra distractor in option E).
• Prime numbers have only two factors: 1 and the number itself.
• Composite numbers have additional factors besides 1 and the number itself.
• We must identify the single composite (non-prime) number among the options.

Concept / Approach:
To check if a number is prime, we test divisibility by prime numbers less than or equal to the square root of the number. If we find any divisor other than 1 and the number itself, the number is composite. For two-digit numbers, testing divisibility by 2, 3, 5, 7, 11 and sometimes 13 is usually enough. We go option by option and see which one can be factored into smaller integers greater than 1.

Step-by-Step Solution:
Step 1: Check 91. It is odd, so not divisible by 2. Sum of digits is 9 + 1 = 10, which is not divisible by 3, so 91 is not divisible by 3. It does not end in 0 or 5, so it is not divisible by 5. Step 2: Test divisibility by 7. Compute 7 × 13 = 91, so 91 = 7 * 13. Therefore 91 has divisors 7 and 13 in addition to 1 and 91 and is composite. Step 3: Check 71. It is not even, digit sum 7 + 1 = 8 is not divisible by 3, it does not end in 0 or 5, and simple checks show it is not divisible by 7 or 11. So 71 is prime. Step 4: Check 41. As with 71, basic tests for 2, 3, 5 and 7 show no divisors. It is a known prime. Step 5: Check 31. It is also not divisible by 2, 3, 5 or 7 and is also a known prime.

Verification / Alternative check:
You can verify that 91 is composite by directly writing 91 ÷ 7 = 13 or 91 ÷ 13 = 7, giving exact integer results. For the others, quick mental checks show no such factorisation. For instance, 71 ÷ 7 is a little more than 10, not an integer, and 71 ÷ 11 is not an integer either. Since only 91 factorises nicely into a product of two smaller natural numbers greater than 1, 91 is the unique non-prime here.

Why Other Options Are Wrong:
Option 71: This is a prime number, as it has no divisor other than 1 and 71 in the tested range.
Option 41: This is also a prime, commonly used as an example in basic prime lists.
Option 31: Another prime number, no factors other than 1 and 31.
Option 61: Included as an extra prime distractor; it is also prime.

Common Pitfalls:
Some students guess based on “shape” of the number or because it looks composite, rather than performing basic divisibility checks. Others may mistakenly think that if a number is not divisible by 2 or 3, it must be prime, which is not always true. For two-digit numbers, always test divisibility by 2, 3, 5, 7 and 11 at least. Recognising that 91 equals 7 * 13 is a useful pattern to remember for future problems.

Final Answer:
The number that is not prime is 91.