Which of the following numbers is not a prime number?

Introduction / Context:
This question asks you to identify which of the given numbers is not prime. A prime number has exactly two distinct positive divisors: 1 and the number itself. A composite number has additional divisors. We must test each option and find the one that has at least one non trivial factor.

Given Data / Assumptions:
The numbers to test are 731, 227, 347 and 461.We need to decide which is composite (not prime).We can use divisibility tests and trial division up to the square root of each number.

Concept / Approach:
To test whether a number N is prime, it is enough to check divisibility by all prime numbers less than or equal to √N. If none divide N, then N is prime. We will test each candidate, focusing on obvious factors first. If we find a factorization, the number is composite.

Step-by-Step Solution:
Step 1: Test 731. Check divisibility by small primes.Step 2: 731 is not even, and the sum of its digits is 7 + 3 + 1 = 11, not divisible by 3, so 2 and 3 do not divide it. It does not end with 5, so 5 does not divide it.Step 3: Check divisibility by 7: 7 × 100 = 700; 7 × 104 = 728, remainder 3, so 7 is not a factor.Step 4: Check 11: 11 × 60 = 660; 11 × 66 = 726, remainder 5, so 11 is not a factor.Step 5: Check 17: 17 × 40 = 680 and 17 × 3 = 51; 680 + 51 = 731, so 17 × 43 = 731. Thus 731 = 17 × 43 and is composite.Step 6: Now test 227. Try dividing by primes up to about 15 (since √227 is a little less than 16). It is not divisible by 2, 3 (digit sum 2 + 2 + 7 = 11), 5, 7, 11 or 13. Hence 227 is prime.Step 7: Test 347. Its square root is less than 19. It is not divisible by 2, 3 (digit sum 3 + 4 + 7 = 14), 5, 7, 11, 13 or 17. So 347 is prime.Step 8: Test 461. Its square root is a little more than 21. It is not divisible by 2, 3 (digit sum 4 + 6 + 1 = 11), 5, 7, 11, 13, 17 or 19. Thus 461 is prime.

Verification / Alternative check:
Once we find that 731 = 17 × 43, we have a clear composite factorization. For the other numbers, systematic checking up to the square root shows no factors. This confirms that 227, 347 and 461 each have no non trivial divisors and are prime.

Why Other Options Are Wrong:
Options (b), (c) and (d) are primes, not composites. They do not have any divisors other than 1 and themselves. Therefore they are not correct answers to the question asking for a non prime number.

Common Pitfalls:
Students sometimes stop testing too early or skip some primes, accidentally labelling a prime as composite or vice versa. Another mistake is to rely solely on divisibility by 2, 3 or 5 without checking higher primes. Using a systematic trial division up to the square root ensures accuracy.

Final Answer:
The number that is not prime is 731, since 731 = 17 × 43.