Bunuel wrote:
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 \(b\)?
A. 12
B. 16
C. 20
D. 30
E. 36
Given:- f(b)*g(a)=0 and f(a)=1.
f(a)=1 means that there is an only even factor for 'a' and that even factor will be 2. Now, either f(b)=0 or g(a)=0 which means either b has 0 even factors or a has 0 odd factors.
So, when b has 0 even factors, least common multiple would be of odd factors. i.e. it would be odd. No such option given.
When a has 0 odd factors, least common multiple would be of at least 1 even factor because f(a)=1, That culminates into that least common multiple would be a factor of 2.
Now, say, lcm is a multiple of 4, then f(a) has to be 2, whereas f(a) is only 1. The result being that required lcm is a product of 2 and other odd factors. The only option that fits the criteria is 30=2*5*3. So, 30, Option D is our answer.