Problem Analysis #25

To solve the problem, we are given a positive integer $n$, and we define:

• $f(n)$: the number of positive divisors of $n$ congruent to 1 modulo 3,
• $g(n)$: the number of positive divisors of $n$ congruent to 2 modulo 3.

We are told that $g(n) = 15$, and the possible values of $f(n)$ are among the following:

$$15,\ 16,\ 18,\ 20,\ 30.$$

Our task is to sum all these possible values of $f(n)$ for which $g(n) = 15$.

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Step 1: Understand the Structure of Divisors Modulo 3

Any positive integer $n$ can be written as:

$$n = 3^k \cdot m,$$

where $m$ is not divisible by 3 (i.e., $\gcd(m, 3) = 1$).

The divisors of $n$ are of the form 3^t $$\cdot$$ d, where 0 $$\leq$$ t $$\leq$$ k and $d \mid m$.

• If $t = 0$, then d $$\equiv$$ 1 or $2 \mod 3$, since we're working in $m$, which is coprime to 3.
• If t $$\geq$$ 1, then 3^t $$\cdot$$ d $$\equiv$$ 0 \mod 3.

Thus, only the divisors of $m$ (those with $t = 0$) contribute to $f(n)$ and $g(n)$.

Let us denote:

• $T$: the number of divisors of $m$ that are $$\equiv$$ 1 \mod 3,
• $Y$: the number of divisors of $m$ that are $$\equiv$$ 2 \mod 3.

Then:

$$f(n) = T \cdot X, \quad g(n) = T \cdot Y,$$

where $X$ is the number of divisors of the non-3 part (i.e., the part coprime to 3) that are $$\equiv$$ 1 \mod 3, and $Y$ is the same for $$\equiv$$ 2 \mod 3.

Given that $g(n) = 15$, we have:

$$T \cdot Y = 15.$$

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Step 2: Analyze Possible Values of $f(n)$

We know:

$$f(n) = T \cdot X, \quad g(n) = T \cdot Y = 15.$$

There are two cases depending on the structure of the type B primes (primes $$\equiv$$ 2 \mod 3) in $m$:

Case A: All type B exponents are even

• Then, for each such prime, $X_i = Y_i + 1$,
• So overall $X = Y + 1$,
• Therefore, f(n) = T $$\cdot$$ (Y + 1) = T $$\cdot$$ Y + T = 15 + T.

Case B: At least one type B exponent is odd

• Then, $X = Y$,
• So f(n) = T $$\cdot$$ Y = 15.

Hence, the possible values of $f(n)$ are:

• $f(n) = 15$ (from Case B),
• $f(n) = 15 + T$ for each divisor $T$ of 15 (from Case A).

Divisors of 15 are $\{1, 3, 5, 15\}$, so the corresponding values of $f(n)$ in Case A are:

• $15 + 1 = 16$,
• $15 + 3 = 18$,
• $15 + 5 = 20$,
• $15 + 15 = 30$.

Thus, the possible values of $f(n)$ when $g(n) = 15$ are:

$$15,\ 16,\ 18,\ 20,\ 30.$$

These are exactly the values listed in the problem statement.

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Step 3: Compute the Sum

Now, we compute the sum of the possible values:

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

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Final Answer

$$\boxed{99}$$