Question

The total number of factors of the square of a prime number is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Always use the formula for number of factors using prime exponent form. For \( p^n \), where \( p \) is prime, total factors = \( n + 1 \).

Answer 1

The Correct Option is C

Step 1: Let the given prime number be \( p \).
Step 2: The square of the prime number is: \[ p^2 \] Step 3: Now find the total number of factors of a number.
If a number is expressed in the form \( n = p_1^{a_1} \cdot p_2^{a_2} \cdot \ldots \cdot p_k^{a_k} \), then the total number of factors is: \[ \text{Total Factors} = (a_1 + 1)(a_2 + 1) \ldots (a_k + 1) \] Step 4: In our case, \( p^2 \) is of the form \( p^2 \), which means: \[ \text{Total Factors} = (2 + 1) = 3 \] Step 5: Therefore, the three factors of \( p^2 \) are: \[ 1, \quad p, \quad p^2 \] \[ \Rightarrow \boxed{\text{Total number of factors is 3}} \]