Problem 3

For any positive integer

n

, let

d(n)

denote the number of positive divisors of

n

(including

1

and

n

itself). Determine all positive integers

k

such that

\frac{d(n^2)}{d(n)}=k

for some

n

Step 1 of 6: Express the divisor ratio

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In plain words

Prime exponents control both divisor counts.

n=\prod_{i=1}^r p_i^{a_i}\quad\Rightarrow\quad \frac{d(n^2)}{d(n)}=\prod_{i=1}^r\frac{2a_i+1}{a_i+1}

Detailed analysis

For the prime factorization $n=\prod p_i^{a_i}$ , the divisor formula gives $d(n)=\prod(a_i+1)$ and $d(n^2)=\prod(2a_i+1)$ . Hence the ratio is the displayed product. Each factor is rational, but the whole product is the integer $k$ in the problem.