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

The function satisfies the equation:

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

Now, let's factorize 160000 into prime factors:

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

Using the given functional equation, we can calculate: $f(160000)$. The equation is:

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

We can further break down $f(2^6)$ and $f(5^5)$ using the functional equation:

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

$f(5^5) = f(5) \cdot f(5^4) + f(5) + f(5^4)$

We can continue this process until we express $f(160000)$ in terms of $f(2)$ and $f(5)$. Since 2 and 5 are prime numbers, $f(2) = 1$ and $f(5) = 1$.

After substituting these values and simplifying, we will get: $f(160000) = 4095$.

Therefore, the value of $f(160000)$ is 4095.