Let's analyze the given functional equation: $f(xy) = f(x)f(y) + f(x) + f(y)$.
This equation can be rewritten as:
$f(xy+1) = (f(x)+1)(f(y)+1)$
We are given the number $160000$. Let's find its prime factorization:
$160000 = 2^6 \times 5^5$
We need to find $f(160000)$. Using the functional equation:
$f(xy) = f(x)f(y) + f(x) + f(y)$
Therefore,
$f(160000) = f(2^6 \cdot 5^5) = f(2^6)f(5^5) + f(2^6) + f(5^5)$
We can compute $f(2^6)$ and $f(5^5)$ by applying the rule recursively, starting from $f(2)$ and $f(5)$.
For $f(2^6)$:
$f(2^6) = f(2)f(2^5) + f(2) + f(2^5)$
Similarly, we can compute $f(2^5), f(2^4)$, and so on, down to $f(2)$. The same process applies to powers of 5.
Given that $f(2) = 1$ and $f(5) = 1$, the recursive application of the equation reveals a pattern. Each step involves multiplication and addition of previous results. Repeating this process yields:
$f(2^6) = 63$
$f(5^5) = 65$
Substituting these values back into the equation for $f(160000)$:
$f(160000) = 63 \cdot 65 + 63 + 65 = 4095$
Thus, the value of $f(160000)$ is $\boxed{4095}$.