Let σ(n) be the sum of the positive divisors of an integer n and ϕ(n) be the number of positive integers smaller than n that are coprime to n. If p is a prime number, what is the maximum value σ(p)ϕ(p)?

To find the maximum value of σ(p)ϕ(p), we need to understand the properties of the functions σ(n) and ϕ(n).

Let's start with σ(n), the sum of positive divisors of n. One way to calculate σ(n) is by finding all the divisors of n and adding them up. However, there is a more efficient way to compute σ(n) using the prime factorization of n.

If we have the prime factorization of n as p₁^α₁ * p₂^α₂ * ... * pᵦ^αᵦ, where pᵢ represents distinct prime factors and αᵢ represents their respective exponents, then the sum of divisors σ(n) can be calculated using the following formula:

σ(n) = (p₁^0 + p₁^1 + ... + p₁^α₁) * (p₂^0 + p₂^1 + ... + p₂^α₂) * ... * (pᵦ^0 + pᵦ^1 + ... + pᵦ^αᵦ)

In simpler terms, we add up all the powers of each prime factor from 0 to their respective exponents and multiply the results from each prime factor together.

Now let's consider ϕ(n), which represents the count of positive integers smaller than n that are coprime to n (i.e., they have no common factors other than 1). A useful property of ϕ(n) is that if n = p₁^α₁ * p₂^α₂ * ... * pᵦ^αᵦ, then ϕ(n) can be calculated using the following formula:

ϕ(n) = n * (1 - 1/p₁) * (1 - 1/p₂) * ... * (1 - 1/pᵦ)

So, the value of ϕ(n) is obtained by multiplying n by the product of (1 - 1/pᵢ) for each distinct prime factor pᵢ.

Now, let's consider the prime number p. In this case, p only has one prime factor, which is itself. Hence, its prime factorization is p^1.

Using the formulas mentioned above:

σ(p) = (p^0 + p^1) = 1 + p
ϕ(p) = p * (1 - 1/p) = p - 1

To maximize the value of σ(p)ϕ(p), we need to find the largest possible value of (1 + p)(p - 1). Since p is a prime number, it must be greater than 1. Thus, the largest possible value occurs when p is the largest prime number possible.

For example, if p = 2, then σ(p)ϕ(p) = (1 + 2)(2 - 1) = 3.
If p = 3, then σ(p)ϕ(p) = (1 + 3)(3 - 1) = 8.
If p = 5, then σ(p)ϕ(p) = (1 + 5)(5 - 1) = 24.

Therefore, the maximum value of σ(p)ϕ(p) occurs when p is the largest prime number possible.