Question

A prime number \( p \) divides \( a^2 \) where \( a \) is a positive integer, then

• A. \( p \) divides \( a \)
• B. \( p \) does not divide \( a \)
• C. \( p \) is equal to \( a \)
• D. All of these

Hint:

When a prime number divides the square of a number, it must divide the number itself.

Answer 1

The Correct Option is A

Step 1: Understanding the Divisibility Condition.
We are given that a prime number \( p \) divides \( a^2 \), where \( a \) is a positive integer. This means: \[ p \mid a^2 \quad \text{(i.e., \( p \) divides \( a^2 \))} \] Step 2: Applying the Fundamental Theorem of Arithmetic.
From the properties of prime numbers, if a prime \( p \) divides the square of a number \( a^2 \), then \( p \) must divide \( a \). This is because prime numbers do not divide a number’s square unless they divide the number itself. This can be formally expressed as: \[ p \mid a^2 \implies p \mid a \] Step 3: Conclusion.
Thus, if \( p \) divides \( a^2 \), then \( p \) must divide \( a \).