CAT 2017

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Functional Equations — Multiplicative Functions
• Formula: $$ f(ab) = f(a) \cdot f(b) $$

2. Calculation

We are given that $$f(ab) = f(a) \cdot f(b)$$ holds for all positive integers $$a$$ and $$b$$.

To find $$f(1)$$, substitute $$a = 1$$ and $$b = 1$$ into the functional equation:

$$f(1 \cdot 1) = f(1) \cdot f(1)$$

$$f(1) = [f(1)]^2$$

This gives us a simple quadratic equation in $$f(1)$$:

$$[f(1)]^2 - f(1) = 0$$

$$f(1)\bigl(f(1) - 1\bigr) = 0$$

So the two possible values are:

$$f(1) = 0 \quad \text{or} \quad f(1) = 1$$

Among these two solutions, the largest value is $$f(1) = 1$$. Note that if $$f(1) = 0$$, the functional equation would force $$f(n) = f(n \cdot 1) = f(n) \cdot f(1) = 0$$ for all positive integers $$n$$, making $$f$$ identically zero — a trivial and less interesting solution. The non-trivial solution $$f(1) = 1$$ is consistent with all multiplicative functions (e.g., $$f(n) = n^k$$).

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3. Solution

Answer = 1 ✅

The largest possible value of $$f(1)$$ is 1.