Positive integer question...

for which positive integers n is τ(n) odd?

Let n = (p1)^(a1) * ... * (pr)^(ar) be the prime factorization of n into distinct primes p1, ..., pr.

Then, since τ(n) = τ((p1)^(a1)) * ... * τ((pr)^(ar)),

τ(n) is odd
<==> τ((pk)^(ak)) is odd for each k = 1, 2, ..., r
<==> (ak + 1) is odd for each k = 1, 2, ..., r
<==> ak is even for each k = 1, 2, ..., r
<==> n is a perfect square.

I hope this helps!