Question

Let $n(>1)$ be a composite integer such that $\sqrt{n}$ is not an integer. Consider the following statements: A: $n$ has a perfect integer-valued divisor which is greater than 1 and less than $\sqrt{n}$ B: $n$ has a perfect integer-valued divisor which is greater than $\sqrt{n}$ but less than $n$ Then:

• A. Both A and B are false
• B. A is true but B is false
• C. A is false but B is true
• D. Both A and B are true

Hint:

For composite numbers, always check the divisors less than and greater than the square root to evaluate the validity of divisor-related statements.

Answer 1

The Correct Option is D

For a composite integer $n$, it has divisors that lie both greater and smaller than $\sqrt{n}$. The integer divisors greater than 1 but less than $\sqrt{n}$ are valid divisors for statement A, and divisors greater than $\sqrt{n}$ but less than $n$ are valid for statement B. Thus, both A and B are true.