Problem Analysis #25

To solve this problem, we need to understand the relationship between the divisors of $n$ that are congruent to $1$ and $2$ modulo $3$, and how $g(n) = 15$ affects $f(n)$.

1. Understanding $g(n)$:
   $g(n)$ is the count of divisors of $n$ congruent to $2$ modulo $3$. Given $g(n) = 15$, we know there are 15 such divisors.

2. Divisors and Modulo 3:
   Divisors of $n$ can be categorized by their residues modulo $3$:

   • Residue $0$: Divisors divisible by $3$.
   • Residue $1$: Divisors congruent to $1$ modulo $3$.
   • Residue $2$: Divisors congruent to $2$ modulo $3$.

   Let $d_0(n)$, $d_1(n)$, and $d_2(n)$ denote the counts of divisors congruent to $0$, $1$, and $2$ modulo $3$, respectively. We have:

   $$d_0(n) + d_1(n) + d_2(n) = \tau(n) \quad \text{(total number of divisors of \(n\))}.$$

   Since $g(n) = d_2(n) = 15$, we know $d_2(n) = 15$.

3. Key Insight:
   For any divisor $d$ of $n$, if $d \equiv 1 \pmod{3}$, then $n/d \equiv 2 \pmod{3}$. Similarly, if $d \equiv 2 \pmod{3}$, then $n/d \equiv 1 \pmod{3}$. This implies that divisors congruent to $1$ and $2$ modulo $3$ come in pairs $(d, n/d)$.

4. Implications of $g(n) = 15$:
   Since $d_2(n) = 15$, there are 15 divisors congruent to $2$ modulo $3$. Each such divisor $d$ has a corresponding divisor $n/d$ congruent to $1$ modulo $3$. Thus, $d_1(n)$ must be at least $15$. However, $d_1(n)$ can be greater than $15$ if there are additional divisors congruent to $1$ modulo $3$ that are not paired with divisors congruent to $2$ modulo $3$.

5. Possible Values of $f(n)$:
   The problem states the possible values of $f(n)$ are $15, 16, 18, 20, 30$. These values represent the number of divisors congruent to $1$ modulo $3$. Since $g(n) = 15$, the minimum $f(n)$ is $15$, but $f(n)$ can be any of the listed values.

6. Summing the Possible Values:
   The sum of all possible values of $f(n)$ is:

   $$15 + 16 + 18 + 20 + 30 = 99.$$

Final Answer: