Tau Tau?

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

[1][1]. The number of integer solutions (x,y)(x, y) to 1x+1y=1n\frac{1}{x}+\frac{1}{y}=\frac{1}{n} is 2τ(n2)2\tau (n^2) where nn is a positive integer.

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

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

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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