For integers $$m$$ and $$n$$, both greater than 1, consider the following three statements : P : m divides n Q : m divides $$n^2$$ R : m is prime, then

We are given three statements for integers $$m$$ and $$n$$, both greater than 1:

• P: $$m$$ divides $$n$$ (i.e., $$n$$ is divisible by $$m$$, written as $$m \mid n$$)
• Q: $$m$$ divides $$n^2$$ (i.e., $$m \mid n^2$$)
• R: $$m$$ is prime

We need to evaluate the logical implications given in the options and determine which one is always true.

Starting with Option A: $$Q \wedge R \rightarrow P$$. This means if Q and R are both true, then P must be true. That is, if $$m$$ divides $$n^2$$ and $$m$$ is prime, then $$m$$ divides $$n$$.

Assume Q and R are true: $$m \mid n^2$$ and $$m$$ is prime. Since $$m$$ is prime and divides $$n^2$$, and $$n^2 = n \times n$$, by Euclid's lemma (which states that if a prime divides a product, it must divide at least one factor), $$m$$ must divide $$n$$. Therefore, P is true. This implication holds.

Now, Option B: $$P \wedge Q \rightarrow R$$. This means if P and Q are true, then R must be true. That is, if $$m$$ divides $$n$$ and $$m$$ divides $$n^2$$, then $$m$$ is prime.

Note that if P is true ($$m \mid n$$), then Q ($$m \mid n^2$$) is automatically true because if $$n = m \cdot k$$ for some integer $$k$$, then $$n^2 = m \cdot (m \cdot k^2)$$, so $$m \mid n^2$$. Thus, the premise $$P \wedge Q$$ reduces to P being true. But the conclusion requires $$m$$ to be prime, which may not hold. For example, let $$m = 4$$ (composite) and $$n = 4$$. Then P: $$4 \mid 4$$ is true (since $$4 = 4 \times 1$$), and Q: $$4 \mid 16$$ is true (since $$16 = 4 \times 4$$), but R: $$m = 4$$ is not prime. Since the premise is true and the conclusion is false, this implication is not always true.

Option C: $$Q \rightarrow R$$. This means if Q is true, then R must be true. That is, if $$m$$ divides $$n^2$$, then $$m$$ is prime.

This is not always true. For example, let $$m = 4$$ and $$n = 2$$. Then Q: $$4 \mid 2^2 = 4$$ is true (since $$4 = 4 \times 1$$), but R: $$m = 4$$ is not prime. Thus, this implication is false.

Option D: $$Q \rightarrow P$$. This means if Q is true, then P must be true. That is, if $$m$$ divides $$n^2$$, then $$m$$ divides $$n$$.

This is not always true. For example, let $$m = 4$$ and $$n = 2$$. Then Q: $$4 \mid 2^2 = 4$$ is true, but P: $$4 \mid 2$$ is false (since $$2 \div 4 = 0.5$$, not an integer). Another example: $$m = 9$$ and $$n = 3$$. Then Q: $$9 \mid 3^2 = 9$$ is true (since $$9 = 9 \times 1$$), but P: $$9 \mid 3$$ is false. Hence, this implication is false.

Only Option A is always true. Therefore, the correct answer is Option A.

Hence, the correct answer is Option A.