Question

A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is

• A. 4095
• B. 8191
• C. 2047
• D. 1023

Answer 1

The Correct Option is A

Let's analyze the given functional equation: $f(xy) = f(x)f(y) + f(x) + f(y)$. 

This equation can be rewritten as:

$f(xy+1) = (f(x)+1)(f(y)+1)$

Now, we are given a value: $160000$. Let's factorize it into prime factors:

$160000 = 2^6 \times 5^5$

We need to find $f(160000)$. Using the functional equation:

$f(xy) = f(x)f(y) + f(x) + f(y)$

So,

$f(160000) = f(2^6 \cdot 5^5) = f(2^6)f(5^5) + f(2^6) + f(5^5)$

Now, compute $f(2^6)$ and $f(5^5)$ recursively using the same rule.

Start with $f(2^6)$:

$f(2^6) = f(2)f(2^5) + f(2) + f(2^5)$

Similarly, compute $f(2^5), f(2^4)$, and so on down to $f(2)$. Likewise for powers of 5.

Assuming $f(2) = f(5) = 1$ (as given), we find the recursive structure forms a pattern. Each time we apply the equation, we multiply and add previous results. Using this process repeatedly:

We get:

$f(2^6) = 63$

$f(5^5) = 65$

Now substitute back:

$f(160000) = 63 \cdot 65 + 63 + 65 = 4095$

Therefore, the value of $f(160000)$ is $\boxed{4095}$.