Question

Let \( A = \{10, 11, 12, 14, 26\} \) and let \( f : A \to \mathbb{N} \) be such that \( f(a) = \) highest prime factor of \( a \), where \( a \in A \), then range of \( f = \)

• A. \( \{5, 7, 13\} \)
• B. \( \{5, 7, 11, 13\} \)
• C. \( \{3, 5, 7, 11, 13\} \)
• D. \( \{3, 7, 11, 13\} \)

Hint:

Highest prime factor of a prime number is the number itself. For composite numbers, factorize completely and pick the largest prime.

Answer 1

The Correct Option is C

Step 1: Understand the function.
The set is $A = \{10, 11, 12, 14, 26\}$. For each number, $f$ gives its highest prime factor. The range is the set of all output values.
Step 2: Factor 10.
$10 = 2\times5$. The highest prime factor is $5$.
Step 3: Factor 11.
$11$ is itself a prime number, so its highest prime factor is $11$.
Step 4: Factor 12.
$12 = 2\times2\times3$. The highest prime factor is $3$.
Step 5: Factor 14 and 26.
$14 = 2\times7$, so the highest prime factor is $7$. $26 = 2\times13$, so the highest prime factor is $13$.
Step 6: Collect the outputs.
The outputs are $5, 11, 3, 7, 13$. As a set the range is $\{3, 5, 7, 11, 13\}$. \[ \boxed{\text{Range} = \{3, 5, 7, 11, 13\}} \]