Question

Let n be the positive integer that has exactly five factors. What is the greatest value of k such that sᵏ?

Answer 1

The greatest value of k such that
`\(s^k\)` represents a positive integer with exactly five factors is 2, corresponding to a perfect square, where
`\(n = p^2\)`, with p being a prime number.

The number of factors of a positive integer n can be determined using its prime factorization. If n has a prime factorization of
`\(p_1^(a_1) * p_2^(a_2) * \ldots * p_k^(a_k)\)`, where
`\(p_i\)` are prime numbers and
`\(a_i\)` are their respective powers, then the total number of factors is given by
`\((a_1 + 1) * (a_2 + 1) * \ldots * (a_k + 1)\)`.

Since we want n to have exactly five factors, there are two possibilities:

1. n is a perfect square, and
`\(a_1\)` is even while all other
`\(a_i\)` are zero.

2. n is the product of two distinct prime numbers.

Now, let's find the greatest value of k such that
`\(s^k\)` represents one of these cases.

1. Case 1 (Perfect square):

Let
`\(n = p^2\)`, where p is a prime number. In this case, k = 2 is the greatest value.

2. Case 2 (Product of two distinct primes):

Let
`\(n = p_1 * p_2\)`, where
`\(p_1\) and \(p_2\)` are distinct prime numbers. In this case, k = 1 is the greatest value.

Therefore, the greatest value of k is 2 for the perfect square case.