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

Options

A. `Q ^^ R to P`
B. `P ^^ Q to R`
C. `Q to R`
D. `Q to P`

Explanation

To analyze the statements given in the question, we need to understand the implications of each statement and how they relate to one another. Let's denote the statements as follows:

- P: m divides n (m | n)
- Q: m divides n² (m | n²)
- R: m is prime

We want to determine the logical relationships among these statements.

### Step 1: Understanding the implications of the statements

1. **If m divides n (P)**, then we can express n as:
\[ n = k \cdot m \quad \text{for some integer } k \]

2. **Now, if m divides n, we can analyze Q**:
\[ n^2 = (k \cdot m)^2 = k^2 \cdot m^2 \]
Since \(m^2\) is clearly divisible by \(m\), it follows that:
\[ m \text{ divides } n^2 \quad \text{(Q is true)} \]
Therefore, if P is true, Q must also be true.

### Step 2: Analyzing the relationship between Q and R

3. **Now consider R**: If m is prime, then it can only divide n or n² if n is a multiple of m. However, the converse is not necessarily true. Just because m divides n² does not imply that m is prime.

### Step 3: Exploring the implications of Q

4. **If m divides n² (Q)**, we can analyze whether this implies P or R:
- If \(m\) divides \(n^2\), it does not necessarily mean that \(m\) divides \(n\). For example, if \(m = 4\) and \(n = 2\), then \(4\) divides \(2^2 = 4\) (Q is true) but \(4\) does not divide \(2\) (P is false).
- Therefore, Q does not imply P.

### Step 4: Exploring the implications of R

5. **If m is prime (R)**, we need to check if it implies P or Q:
- If \(m\) is prime and divides \(n\), then it must also divide \(n^2\) since \(n^2\) is a product of \(n\) with itself. Thus, if R is true, then both P and Q are true.

### Conclusion

From the analysis above, we can summarize the implications:
- If P is true, then Q is true.
- If R is true, then both P and Q are true.
- Q does not necessarily imply P or R.

Thus, the correct logical relationship among the statements is:
- Q and R imply P.

### Final Answer

The correct option is: **Q and R conditional P**.

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