The function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b?  Choose the correct answer.
A) 12
B) 16
C) 20
D) 30
E) 36
f(b)∗g(a)=0 : any positive integer has at least one odd factor: 1. Thus, g(a) cannot be 0, which implies that f(b) = 0. This on the other hand means that b is an odd integer (odd integers does not have even factors).   f(a)=1f(a)=1: aa has 1 even factor. This means that aa must be 2 (the only positive integer which has only 1 even factor is 2).   The least common multiple of a=2a=2 and b=odd is 2∗odd. Only option D can be represented this way: 30=2∗1530=2∗15.   Answer: 30
<result>D</result>
The function f(x) is defined for all positive integers x as the number of even factors of x and the function g(x) is defined for all positive integers x as the number of odd factors of x. For positive integers a and b if f(b)*g(a) = 0 and f(a) = 1, which of the following could be the least common multiple of a and b?