Question

Let f be a function such that f (mn) = f (m) f (n) for every positive integers m and n. If f (1), f (2) and f (3) are positive integers, f (1) < f (2), and f (24) = 54, then f (18) equals

• A. 12
• B. 14
• C. 17
• D. 22

Answer 1

The Correct Option is A

We are given a function \( f \) such that: \[ f(mn) = f(m) \cdot f(n) \] for all positive integers \( m \) and \( n \).

Additional conditions:

• \( f(1) < f(2) \)
• \( f(1), f(2), f(3) \) are positive integers
• \( f(24) = 54 \)

Step 1: Find \( f(1) \)

Using the multiplicative identity: \[ f(1) = f(1 \cdot 1) = f(1)^2 \Rightarrow f(1)^2 = f(1) \Rightarrow f(1) = 1 \quad (\text{since } f(1) > 0) \]

Step 2: Use Known Values to Form an Equation

Factor 24: \( 24 = 2^3 \cdot 3 \) \[ f(24) = f(2^3 \cdot 3) = f(2)^3 \cdot f(3) \] Let \( f(2) = a \) and \( f(3) = b \). Then: \[ a^3 \cdot b = 54 \]

Step 3: Find Integer Solutions for \( a^3 \cdot b = 54 \)

Factor 54: \( 54 = 2 \cdot 3^3 \) Try \( a = 3 \Rightarrow a^3 = 27 \Rightarrow b = \frac{54}{27} = 2 \)
This gives valid positive integers: \( a = 3, b = 2 \)

Step 4: Compute \( f(18) \)

Factor 18: \( 18 = 2 \cdot 3^2 \) \[ f(18) = f(2) \cdot f(3)^2 = a \cdot b^2 = 3 \cdot 2^2 = 3 \cdot 4 = 12 \]

Final Answer:

\( f(18) = 12 \)