Question

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(l) is

• A. 1
• B. 3
• C. 5
• D. 4

Answer 1

The Correct Option is A

To determine the largest possible value of \( f(l) \) given that \( f(ab) = f(a)f(b) \) for all positive integers \( a \) and \( b \), we must consider the functional equation and its implications. A function satisfying this equation for all positive integers is known as a multiplicative function. One common example is \( f(x) = x^c \) for a constant \( c \). Let's explore this function: 

• We need to check if such a function can achieve values larger than 1.
• If \( f \) is a constant function, then it must hold that \( f(x) = c \). The equation \( f(ab) = f(a)f(b) \) implies \( c = c \cdot c \), resulting in \( c^2 = c \). The solutions to this equation are \( c = 0 \) or \( c = 1 \).
• Therefore, the function could potentially be \( f(x) = 0 \) or \( f(x) = 1 \). A function like \( f(x) = x^0 = 1 \) satisfies the relation and gives a consistent value of 1 for all \( x \).

Since all multiplicative functions of this type either result in 0 or 1, and the non-zero solution gives us the constant function \( f(x) = 1 \) for any positive integer \( x \), the largest possible value of \( f(l) \) is 1.