If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp...

1. Translate the problem requirements: We need to understand what it means for \(\mathrm{pq}\) (product of two different primes) to divide \(\mathrm{mp + nq}\), and determine which of the three statements must always be true.
2. Apply divisibility logic to the given condition: Since \(\mathrm{pq}\) divides \(\mathrm{mp + nq}\), we can write \(\mathrm{mp + nq = k(pq)}\) for some integer \(\mathrm{k}\), then rearrange to isolate terms.
3. Analyze what each prime must divide: Use the fact that if a prime divides a sum, and it divides one term, it must also divide the other term to find necessary conditions.
4. Test each statement systematically: Check whether each of the three given statements (I, II, III) must be true based on our divisibility analysis.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given in everyday language:

• \(\mathrm{p}\) and \(\mathrm{q}\) are different prime numbers (like 2, 3, 5, 7, 11, etc.)
• \(\mathrm{m}\) and \(\mathrm{n}\) are integers (any whole numbers, positive, negative, or zero)
• \(\mathrm{pq}\) divides \(\mathrm{mp + nq}\) (this means \(\mathrm{mp + nq}\) is a multiple of \(\mathrm{pq}\))

We need to figure out which of three statements must ALWAYS be true when these conditions hold.

2. Apply divisibility logic to the given condition

Since \(\mathrm{pq}\) divides \(\mathrm{mp + nq}\), this means we can write:

\(\mathrm{mp + nq = k \times pq}\) for some integer \(\mathrm{k}\)

Let's rearrange this equation to see what it tells us:

\(\mathrm{mp + nq = kpq}\)

\(\mathrm{mp + nq = kpq + 0}\)

\(\mathrm{mp - kpq = -nq}\)

\(\mathrm{p(m - kq) = -nq}\)

This tells us that \(\mathrm{p}\) times something equals \(\mathrm{-nq}\), which means \(\mathrm{p}\) divides \(\mathrm{nq}\).

3. Analyze what each prime must divide

Now here's the key insight: since \(\mathrm{p}\) and \(\mathrm{q}\) are different primes, and \(\mathrm{p}\) divides \(\mathrm{nq}\), what must \(\mathrm{p}\) divide?

Since primes have no factors other than 1 and themselves, and \(\mathrm{p \neq q}\) (they're different primes), \(\mathrm{p}\) cannot divide \(\mathrm{q}\). Therefore, \(\mathrm{p}\) must divide \(\mathrm{n}\).

Similarly, let's look at what \(\mathrm{q}\) divides. From our equation \(\mathrm{p(m - kq) = -nq}\), we can also write:

\(\mathrm{mp + nq = kpq}\)

\(\mathrm{nq + mp = kpq}\)

\(\mathrm{nq - kpq = -mp}\)

\(\mathrm{q(n - kp) = -mp}\)

This means \(\mathrm{q}\) divides \(\mathrm{mp}\). Since \(\mathrm{q \neq p}\) (different primes), \(\mathrm{q}\) cannot divide \(\mathrm{p}\), so \(\mathrm{q}\) must divide \(\mathrm{m}\).

4. Test each statement systematically

Now let's check each statement:

Statement I: \(\mathrm{p}\) is a divisor of \(\mathrm{n}\)

From our analysis above, we showed that \(\mathrm{p}\) must divide \(\mathrm{n}\). This is ALWAYS true. ✓

Statement II: \(\mathrm{pq}\) is a divisor of \(\mathrm{mp}\)

We know that \(\mathrm{q}\) divides \(\mathrm{m}\) and \(\mathrm{p}\) divides \(\mathrm{p}\) (obviously). Therefore \(\mathrm{pq}\) divides \(\mathrm{mp}\). This is ALWAYS true. ✓

Statement III: \(\mathrm{p^2}\) is a divisor of \(\mathrm{mn}\)

We know \(\mathrm{p}\) divides \(\mathrm{n}\) and \(\mathrm{q}\) divides \(\mathrm{m}\), so \(\mathrm{pq}\) divides \(\mathrm{mn}\). But this doesn't mean \(\mathrm{p^2}\) divides \(\mathrm{mn}\). For example, if \(\mathrm{p = 2}\), \(\mathrm{n = 6}\), then \(\mathrm{p}\) divides \(\mathrm{n}\), but \(\mathrm{p^2 = 4}\) doesn't necessarily divide \(\mathrm{mn}\) unless there are additional factors of \(\mathrm{p}\). This is NOT always true. ✗

5. Final Answer

Statements I and II must be true, while statement III is not necessarily true.

Looking at our answer choices, "I and II" corresponds to choice D.

Answer: D

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "must be true" means: Students often confuse "must be true" with "could be true." They might think that finding one example where a statement works is sufficient, rather than proving the statement is always true under the given conditions.

2. Overlooking the constraint that \(\mathrm{p}\) and \(\mathrm{q}\) are different primes: Students may forget this crucial detail and allow \(\mathrm{p = q}\) in their analysis, which leads to incorrect conclusions about what each prime must divide.

3. Not recognizing the need to use prime factorization properties: Students might try to work with specific numbers or use algebraic manipulation without leveraging the fundamental property that primes can only divide a product if they divide one of the factors.

Errors while executing the approach

1. Algebra manipulation errors when rearranging the divisibility condition: When transforming "\(\mathrm{pq}\) divides \(\mathrm{mp + nq}\)" into useful forms like \(\mathrm{p(m - kq) = -nq}\), students often make sign errors or incorrectly factor expressions.

2. Incorrectly applying the fundamental property of primes: Students might conclude that since \(\mathrm{p}\) divides \(\mathrm{nq}\), then \(\mathrm{p}\) divides both \(\mathrm{n}\) and \(\mathrm{q}\), forgetting that \(\mathrm{p}\) only needs to divide one of the factors since \(\mathrm{p}\) and \(\mathrm{q}\) are different primes.

3. Confusing "\(\mathrm{pq}\) divides \(\mathrm{mn}\)" with "\(\mathrm{p^2}\) divides \(\mathrm{mn}\)": When checking Statement III, students might think that because \(\mathrm{p}\) divides \(\mathrm{n}\) and \(\mathrm{q}\) divides \(\mathrm{m}\), this automatically means \(\mathrm{p^2}\) divides \(\mathrm{mn}\), not recognizing that this would require an additional factor of \(\mathrm{p}\).

Errors while selecting the answer

1. Not systematically checking all three statements: Students might stop after finding that Statement I is true and select "A. I only" without thoroughly verifying whether Statement II is also always true.

2. Misreading the Roman numeral combinations: Students might correctly identify that Statements I and II are true but accidentally select the wrong answer choice due to confusion with the Roman numeral format or not carefully matching their conclusions to the given options.