Question

If \(7\) divides \(a^2\) then

• A. 7 divides \(a\)
• B. 7 divides \(\sqrt a\)
• C. \(a\) divides 7
• D. None

Answer 1

The Correct Option is A

To solve this problem, we need to determine the relationship between \( 7 \) and \( a \) when we know that \( 7 \) divides \( a^2 \).

1. The Condition Given:
The problem states that \( 7 \) divides \( a^2 \), i.e., \( 7 \mid a^2 \). This means that \( a^2 \) is divisible by \( 7 \). According to number theory, if a prime \( p \) divides the square of a number, then \( p \) must divide the number itself.

2. Applying the Rule:
Since \( 7 \mid a^2 \), we can conclude that \( 7 \) must also divide \( a \) (i.e., \( 7 \mid a \)). This is because if a prime divides the square of a number, it divides the number itself.

3. Evaluating the Options:

• (A) \( 7 \) divides \( a \) – Correct ✔; as explained, if \( 7 \mid a^2 \), then \( 7 \mid a \).
• (B) \( 7 \) divides \( \sqrt{a} \) – Incorrect; we cannot conclude that \( 7 \) divides \( \sqrt{a} \) from the given condition.
• (C) \( a \) divides \( 7 \) – Incorrect; the condition does not imply this.
• (D) None – Incorrect; the correct answer is option (A).

Final Answer:
The correct answer is (A) 7 divides \( a \).