My prof. said this would be the hardest prob on my hw .

1a.) For all integers n >= 1, prove there exists integers a and b such that tau(a) + tau(b) = n, where the number of divisors func., denoted by tau, is defined by setting tau(n) equal to the # of pos. divisors of n.

b.) Now prove the Goldbach conjecture says that for each even integer 2n, there exists integers a and b such that sigma(a) + sigma(b) = 2n, where the sum of the divisors func., denoted by sigma, is defined by setting sigma(n) equal to the sum of all pos. divisors of n.

(Note that tau/sigma are multiplicative functions).