Final answer:
The number with the smallest non-one factor larger than the smallest non-one factor of all smaller numbers is a prime number, as prime numbers only have two distinct divisors: 1 and themselves.
Step-by-step explanation:
The number described is seeking the smallest non-one factor that is larger than the smallest non-one factor of all smaller numbers. The correct answer would be a prime number, option (a). Prime numbers are those that have exactly two distinct positive divisors: 1 and the number itself. Thus, the smallest non-one factor of a prime number is the number itself, which is larger than the smallest non-one factor of any smaller number. Composite numbers, perfect numbers, and square numbers usually have smaller factors, and thus do not fit the description provided.