CAT 2024 Slot 2 QA Question 9

Multiple choice (+3 / −1) · Algebra · Functions · Try it, then check the answer and solution below.

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)

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Official Convenor Answer Key: C

Answer and solution

Answer: C) 4095

📌 Core Concept & Formula

The function

f

satisfies the property

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

for all natural numbers

x

and

y

. By defining a new function

g(x) = f(x) + 1 , we transform this into a multiplicative function:

g(xy) = g(x)g(y)

. For any prime

p , g(p) = 2$.

🔢 Step-by-Step Solution

Define the Function

g

: Let

g(x) = f(x) + 1

. Then,

f(xy) + 1 = (f(x) + 1)(f(y) + 1) , so

g(xy) = g(x)g(y)$.

Prime Values: For any prime

p , f(p) = 1

implies

g(p) = 2

Factorize 160000: 160000 = 2^8 × 5^4$.

Compute

g(160000)

: Since

g

is multiplicative, \( g(1600

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