# Tau Tau?

$$\tau (n)$$ denotes the number of positive divisors a positive integer $$n$$ has. Keeping that in mind, read the following statements below.

$$[1]$$. The number of integer solutions $$(x, y)$$ to $$\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$$ is $$2\tau (n^2)$$ where $$n$$ is a positive integer.

$$[2]$$. $$\tau (n)$$ can never be an odd number.

$$[3]$$. $$\tau (n)$$ is always strictly less than $$n$$.

Which of these statements are correct?

Note: This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.

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