Question

If \( m \) and \( n \) are positive integers, is \( \sqrt{m} \times \sqrt{n} \) an integer?
(1) \( \sqrt{m} \) is an integer.
(2) \( \sqrt{n} \) is an integer.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When working with square roots, remember that for a product of square roots to be an integer, the product inside the square root must be a perfect square.

Answer 1

The Correct Option is D

Step 1: Analyze statement (1).
Statement (1) tells us that \( \sqrt{m} \) is an integer, which implies that \( m \) is a perfect square. Let \( m = a^2 \) for some integer \( a \). Then, we have: \[ \sqrt{m} = a \] Now, consider the expression \( \sqrt{m} \times \sqrt{n} \): \[ \sqrt{m} \times \sqrt{n} = \sqrt{mn} \] For this to be an integer, \( mn \) must be a perfect square. However, we do not know whether \( n \) is a perfect square or not, so statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that \( \sqrt{n} \) is an integer, which implies that \( n \) is a perfect square. Let \( n = b^2 \) for some integer \( b \). Then, we have: \[ \sqrt{n} = b \] Now, consider the expression \( \sqrt{m} \times \sqrt{n} \): \[ \sqrt{m} \times \sqrt{n} = \sqrt{mn} \] For this to be an integer, \( mn \) must be a perfect square. Since \( n \) is a perfect square, we only need to check if \( m \) is a perfect square. Statement (2) alone guarantees that \( \sqrt{m} \times \sqrt{n} \) is an integer if \( m \) is a perfect square. Thus, statement (2) is sufficient. \[ \boxed{D} \]