Final answer:
The correct answer is c) 3137, as it is the smallest positive integer that is neither prime nor square and has no prime factors less than 50. It is the product of two primes greater than 50, specifically 53 and 59.
Step-by-step explanation:
The student has asked for the smallest positive integer that is neither prime nor square and has no prime factor less than 50. To solve for this requirement, we must find a number that is a product of prime factors, each greater than or equal to 50 and is not a perfect square. Here's a step-by-step approach to eliminate the options:
- Option (a) 3127 is a prime number, thus, cannot have any prime factors, and it certainly can't be composed of prime factors all of which are greater than 50. So, it doesn’t meet the criteria.
- Option (b) 3133 is not a prime number. It is 31 times 101, but since 31 is less than 50, it does not meet the criteria.
- Option (c) 3137 is 53 times 59, both of which are primes greater than 50. Additionally, 3137 is not a perfect square, thus it satisfies all criteria, and it is the smallest of the options that does so.
- Option (d) 3139 is a prime number, which means it has no prime factors other than itself, thus not meeting the criteria.
- Option (e) 3149 is also a prime number, which means it has no prime factors other than itself, thus not meeting the criteria.
Therefore, the correct answer is c) 3137, as it is the smallest positive integer that is neither prime nor square and has no prime factors less than 50.