If m and p are positive integers and m^2 + p^2 , what is the greatest possible value of (mp?

1. Translate the problem requirements: We need to find positive integers m and p where the sum of their squares is less than 100, and we want to maximize their product \(\mathrm{mp}\).
2. Establish the search boundary: Determine the maximum possible values for m and p individually based on the constraint \(\mathrm{m}^2 + \mathrm{p}^2 < 100\).
3. Identify promising candidates: Focus on integer pairs that could yield large products while satisfying the constraint, particularly those where both numbers are reasonably large.
4. Test and verify the optimal pair: Check the most promising combinations systematically to find the maximum value of \(\mathrm{mp}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for in everyday language. We have two positive whole numbers, m and p. The constraint tells us that when we square each number and add those squares together, the result must be less than 100. Our goal is to find the pair of numbers that gives us the biggest possible product when we multiply m and p together.

In mathematical terms: We need positive integers m and p such that \(\mathrm{m}^2 + \mathrm{p}^2 < 100\), and we want to maximize \(\mathrm{mp}\).

Process Skill: TRANSLATE - Converting the problem statement into clear mathematical understanding

2. Establish the search boundary

To make our search manageable, let's figure out how big m and p can possibly be. Since both m and p are positive integers and \(\mathrm{m}^2 + \mathrm{p}^2 < 100\), let's think about the maximum value either could have.

If one of the numbers were as large as possible, what would that be? Well, if \(\mathrm{m} = 9\), then \(\mathrm{m}^2 = 81\). For the constraint to work, we'd need \(\mathrm{p}^2 < 100 - 81 = 19\), which means p could be at most 4 (since \(4^2 = 16 < 19\), but \(5^2 = 25 > 19\)).

If \(\mathrm{m} = 10\), then \(\mathrm{m}^2 = 100\), which already violates our constraint since we need the sum to be less than 100.

So both m and p must be 9 or smaller. This gives us a reasonable range to check: both numbers are between 1 and 9.

3. Identify promising candidates

Since we want to maximize the product \(\mathrm{mp}\), we should focus on pairs where both numbers are reasonably large. Small numbers like 1 or 2 won't give us big products, so let's prioritize checking larger pairs first.

Let's think strategically: pairs like (7,7), (6,8), (7,6), (8,6), etc. are good candidates because they involve larger numbers that could give substantial products.

Let me check some promising combinations:

• If \(\mathrm{m} = 7\) and \(\mathrm{p} = 7\): \(\mathrm{m}^2 + \mathrm{p}^2 = 49 + 49 = 98 < 100\) ✓, and \(\mathrm{mp} = 49\)
• If \(\mathrm{m} = 6\) and \(\mathrm{p} = 8\): \(\mathrm{m}^2 + \mathrm{p}^2 = 36 + 64 = 100\), but we need < 100, so this doesn't work ✗
• If \(\mathrm{m} = 7\) and \(\mathrm{p} = 6\): \(\mathrm{m}^2 + \mathrm{p}^2 = 49 + 36 = 85 < 100\) ✓, and \(\mathrm{mp} = 42\)

Process Skill: APPLY CONSTRAINTS - Systematically checking which pairs satisfy our constraint

4. Test and verify the optimal pair

Let me check all the viable combinations systematically, focusing on the larger numbers first:

For \(\mathrm{m} = 7\):

• \(\mathrm{p} = 7\): \(7^2 + 7^2 = 49 + 49 = 98 < 100\) ✓, \(\mathrm{mp} = 49\)
• \(\mathrm{p} = 6\): \(7^2 + 6^2 = 49 + 36 = 85 < 100\) ✓, \(\mathrm{mp} = 42\)
• \(\mathrm{p} = 5\): \(7^2 + 5^2 = 49 + 25 = 74 < 100\) ✓, \(\mathrm{mp} = 35\)

For \(\mathrm{m} = 6\):

• \(\mathrm{p} = 8\): \(6^2 + 8^2 = 36 + 64 = 100 \not< 100\) ✗
• \(\mathrm{p} = 7\): Same as (7,6) above, \(\mathrm{mp} = 42\)
• \(\mathrm{p} = 6\): \(6^2 + 6^2 = 36 + 36 = 72 < 100\) ✓, \(\mathrm{mp} = 36\)

For \(\mathrm{m} = 8\):

• \(\mathrm{p} = 6\): Same as (6,8), doesn't work ✗
• \(\mathrm{p} = 5\): \(8^2 + 5^2 = 64 + 25 = 89 < 100\) ✓, \(\mathrm{mp} = 40\)
• \(\mathrm{p} = 4\): \(8^2 + 4^2 = 64 + 16 = 80 < 100\) ✓, \(\mathrm{mp} = 32\)

The largest product we've found is \(\mathrm{mp} = 49\) when \(\mathrm{m} = 7\) and \(\mathrm{p} = 7\).

Process Skill: CONSIDER ALL CASES - Systematically checking the most promising combinations to ensure we find the maximum

5. Final Answer

The greatest possible value of \(\mathrm{mp}\) is 49, achieved when \(\mathrm{m} = 7\) and \(\mathrm{p} = 7\).

Verification: \(7^2 + 7^2 = 49 + 49 = 98 < 100\) ✓

This matches answer choice D: 49.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the constraint inequality
Students often misread \(\mathrm{m}^2 + \mathrm{p}^2 < 100\) as \(\mathrm{m}^2 + \mathrm{p}^2 \leq 100\). This seemingly small error would allow pairs like (6,8) where \(6^2 + 8^2 = 100\), leading them to consider \(\mathrm{mp} = 48\) as a valid answer when it actually violates the strict inequality requirement.

2. Failing to establish proper search boundaries
Without systematically determining that both m and p must be ≤ 9, students may waste time checking impossible combinations or miss the optimal solution entirely. They might randomly test pairs without understanding the logical limits imposed by the constraint.

3. Adopting an inefficient search strategy
Many students will start checking from small values like (1,1), (1,2), etc., rather than focusing on larger pairs that are more likely to yield maximum products. This approach is time-consuming and may cause them to run out of time before finding the optimal answer.

Errors while executing the approach

1. Arithmetic calculation errors
Students frequently make mistakes when calculating squares and sums. For example, they might compute \(7^2 + 7^2 = 49 + 49 = 96\) instead of 98, or miscalculate products like \(7 \times 7 = 48\) instead of 49. These computational errors can lead to wrong conclusions about which pairs satisfy the constraint.

2. Incomplete systematic checking
Even when students adopt the right approach, they often fail to check all promising combinations. They might find one good pair like (7,6) giving \(\mathrm{mp} = 42\) and stop there, missing the optimal pair (7,7) that gives \(\mathrm{mp} = 49\).

Errors while selecting the answer

1. Selecting a suboptimal but valid solution
Students may find several valid pairs like (8,5) with \(\mathrm{mp} = 40\), (7,6) with \(\mathrm{mp} = 42\), or (6,6) with \(\mathrm{mp} = 36\), and incorrectly select one of these as their final answer instead of continuing to verify they've found the true maximum value of \(\mathrm{mp} = 49\).