Question

The function \( f : \mathbb{N} - \{1\} \to \mathbb{N} \); defined by \( f(n) \) = the highest prime factor of \( n \), is:

• A. both one-one and onto
• B. one-one only
• C. onto only
• D. neither one-one nor onto

Answer 1

The Correct Option is D

Step 1. Understanding the Function \( f(n) \): The function \( f(n) \) maps each natural number \( n \) (excluding 1) to its highest prime factor. For example:
 
 \(f(10) = 5, \quad f(15) = 5, \quad f(18) = 3\)

Step 2. Checking if \( f(n) \) is One-One: For a function to be one-one (injective), each distinct input must map to a distinct output. However, different values of \( n \) can have the same highest prime factor. For instance:

\(f(10) = f(15) = 5\)

  - Since different numbers can yield the same highest prime factor, \( f(n) \) is not one-one.

Step 3. Checking if \( f(n) \) is Onto: For \( f(n) \) to be onto (surjective), every natural number should appear as an output of \( f(n) \). However, not all natural numbers are prime. Since \( f(n) \) only outputs prime numbers, it cannot cover all natural numbers. Therefore, \( f(n) \) is not onto.

Since \( f(n) \) is neither one-one nor onto, the correct answer is \( (4) \).