If $$ f $$ 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

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Accepted Answer

We are tasked with finding $$ f(160000) $$ for a function $$ f $$ that satisfies the following properties:

1. $$ f(xy) = f(x)f(y) + f(x) + f(y) $$ for all $$ x, y $$.
2. $$ f(p) = 1 $$ for every prime number $$ p $$.

#### Step 1: Simplify the functional equation
Rewriting the given equation:
$$
f(xy) = (f(x) + 1)(f(y) + 1) - 1
$$
This form will simplify calculations.

#### Step 2: Break down $$ 160000 $$
We can write:
$$
160000 = 400 	imes 400
$$

Using the functional equation:
$$
f(160000) = (f(400) + 1)(f(400) + 1) - 1
$$

#### Step 3: Calculate $$ f(400) $$
Similarly, write:
$$
400 = 20 	imes 20
$$

Using the functional equation:
$$
f(400) = (f(20) + 1)(f(20) + 1) - 1
$$

#### Step 4: Calculate $$ f(20) $$
Write:
$$
20 = 4 	imes 5
$$

Using the functional equation:
$$
f(20) = (f(4) + 1)(f(5) + 1) - 1
$$

#### Step 5: Calculate $$ f(4) $$
Write:
$$
4 = 2 	imes 2
$$

Using the functional equation and knowing that $$ f(2) = 1 $$, since $$ 2 $$ is a prime number:
$$
f(4) = (f(2) + 1)(f(2) + 1) - 1
$$
Substitute $$ f(2) = 1 $$:
$$
f(4) = (1 + 1)(1 + 1) - 1 = 3
$$

#### Step 6: Calculate $$ f(20) $$
Substitute $$ f(4) = 3 $$, and $$ f(5) = 1 $$, since $$ 5 $$ is a prime number:
$$
f(20) = (3 + 1)(1 + 1) - 1
$$
Simplify:
$$
f(20) = (4)(2) - 1 = 7
$$

#### Step 7: Calculate $$ f(400) $$
Substitute $$ f(20) = 7 $$:
$$
f(400) = (7 + 1)(7 + 1) - 1
$$
Simplify:
$$
f(400) = (8)(8) - 1 = 63
$$

#### Step 8: Calculate $$ f(160000) $$
Substitute $$ f(400) = 63 $$:
$$
f(160000) = (63 + 1)(63 + 1) - 1
$$
Simplify:
$$
f(160000) = (64)(64) - 1 = 4095
$$

### Final Answer:
Option **C:** **4095**

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