For a positive integer \( n \), let \( d(n) \) be the number of positive divisors of \( n \). Prove that, for any positive integer \( M \), there exists a constant \( C>0 \) such that \( d(n) \geq C ( \log n )^M \) for infinitely many \( n \).

Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.

Let n be a positive integer. Let D(n,k) be the number of divisors x of n such that x≡k (mod 3). Prove that D(n,1)≥D(n,2).