Assume that the function d is stated as follows. The domain (input set) is the set of positive intergers, and d(x) is the number of different prime numbers that divide x. For example, d(50)=2, because 50=2x5^2, so 50 has the two prime divisors 2 and 5.
a) For which numbers x is d(x)=1? A complete answer here will give a nice simple characterization of this set of numbers.
b) Find numbers x and y such that d(x+y)=d(x)+d(y). One example is enough here.
c) Find numbers x and y such that d(x+y) does not = d(x)+d(y). One example is enough here.
d) Is there a smallest positive interger for which the output of d is 10 when this number is the input? (In other words, does the set of numbers x for which d(x)=10 have a smallest element?) If so, find it (with justification); if not, explain why not.