Tau Tau?

Number Theory Level 4

\(\tau (n)\) 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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