Number of positive integers `x` for which `f(x)=x^3-8x^2+20 x-13` is a prime number is______.

Explanation

Clearly, `f(1) = 0`
`therefore f(x) = (x - 1) (x^(2) - 7x + 13)`
For f(x) to be prime, at least one of the factors must be prime. Hence, either x - 1 = 1
or `x^(2) - 7x + 13 = 1`
If x - 1 = 1, then x = 2
If `x^(2) - 7x + 13 = 1 ,` then` x^(2) - 7x + 12 = 0`
`rArr x = 3 or 4`
So, x = 2, 3, 4