If p has a total of 17 positive factors, what can be inferred about p?

When considering the total number of positive factors (also known as divisors) of a number, it’s important to understand how that number can be expressed in terms of its prime factorization. If p is expressed as ( p = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} ), where ( p_1, p_2, \ldots, p_k ) are prime factors and ( e_1, e_2, \ldots, e_k ) are their respective powers, the formula for the total number of positive divisors is:

[

(e_1 + 1)(e_2 + 1) \ldots (e_k + 1)

]

Given that p has 17 positive factors, we observe that 17 is a prime number. The only way to achieve a product equal to 17 in the context of divisors, using the formula above, is if there is exactly one prime factor raised to the power of 16. This is because ( 17 = (16 + 1) ), meaning it can only be expressed as ( 17 = 1 \