In a certain deck of cards, each card has a positive integer written on it. In a multiplication game, a...

1. Translate the problem requirements: Each card has a positive integer \(\mathrm{n}\), and when drawn, we multiply \(\mathrm{n}\) by \(\mathrm{n+1}\). We need all possible products \(\mathrm{n(n+1)}\) to fall between 15 and 200 inclusive, and we want to find the smallest and largest possible values of \(\mathrm{n}\) on the cards.
2. Find the minimum integer constraint: Determine the smallest integer \(\mathrm{n}\) where \(\mathrm{n(n+1)} \geq 15\) by testing values around where \(\mathrm{n}^2\) approaches 15.
3. Find the maximum integer constraint: Determine the largest integer \(\mathrm{n}\) where \(\mathrm{n(n+1)} \leq 200\) by testing values around where \(\mathrm{n}^2\) approaches 200.
4. Verify bounds and select answer: Confirm that our minimum and maximum values satisfy the constraints and match one of the given answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this multiplication game. A child draws a card with some positive integer written on it - let's call this integer \(\mathrm{n}\). The rule of the game is simple: take that number and multiply it by the next larger integer, which would be \(\mathrm{n + 1}\).

For example, if a card has the number 5 on it, the child multiplies \(5 \times 6 = 30\). If a card has 8, they calculate \(8 \times 9 = 72\).

The constraint is that every possible product must be between 15 and 200, inclusive. This means we need to find the smallest and largest integers that could be written on the cards such that when we multiply each by its next larger integer, all results fall within this range.

In mathematical terms: we need all products \(\mathrm{n(n+1)}\) to satisfy \(15 \leq \mathrm{n(n+1)} \leq 200\).

Process Skill: TRANSLATE - Converting the game rules into mathematical constraints

2. Find the minimum integer constraint

Now let's find the smallest integer that could be on a card. We need \(\mathrm{n(n+1)} \geq 15\).

Let's test some small positive integers:

• If \(\mathrm{n} = 3\): then \(3 \times 4 = 12\). Since \(12 < 15\), this doesn't work.
• If \(\mathrm{n} = 4\): then \(4 \times 5 = 20\). Since \(20 \geq 15\), this works!

So the smallest integer that can be on a card is 4, because \(4 \times 5 = 20 \geq 15\), but \(3 \times 4 = 12 < 15\).

Process Skill: APPLY CONSTRAINTS - Testing boundary values to find the minimum

3. Find the maximum integer constraint

Now let's find the largest integer that could be on a card. We need \(\mathrm{n(n+1)} \leq 200\).

Since we're looking for larger numbers, let's estimate first. We need \(\mathrm{n}^2\) to be roughly around 200, so \(\mathrm{n}\) should be around \(\sqrt{200} \approx 14.1\). Let's test values around this:

• If \(\mathrm{n} = 13\): then \(13 \times 14 = 182\). Since \(182 \leq 200\), this works.
• If \(\mathrm{n} = 14\): then \(14 \times 15 = 210\). Since \(210 > 200\), this doesn't work.

So the largest integer that can be on a card is 13, because \(13 \times 14 = 182 \leq 200\), but \(14 \times 15 = 210 > 200\).

Process Skill: APPLY CONSTRAINTS - Testing boundary values to find the maximum

4. Verify bounds and select answer

Let's double-check our work:

• Minimum value \(\mathrm{n} = 4\): \(4 \times 5 = 20\), and \(20 \geq 15\) ✓
• Maximum value \(\mathrm{n} = 13\): \(13 \times 14 = 182\), and \(182 \leq 200\) ✓
• Boundary check: \(3 \times 4 = 12 < 15\) (too small) and \(14 \times 15 = 210 > 200\) (too large) ✓

Therefore, the least and greatest integers on the cards could be 4 and 13.

Looking at our answer choices, this matches choice C. 4 and 13.

Final Answer

The correct answer is C. 4 and 13. The smallest integer on any card must be 4 (since \(4 \times 5 = 20 \geq 15\)), and the largest integer on any card can be 13 (since \(13 \times 14 = 182 \leq 200\)).

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "next larger integer"

Students may confuse "next larger integer" with "any larger integer" or "double the integer." For example, if the card shows 5, they might think they need to multiply by 6, 7, 8, or even 10, rather than specifically the immediate next integer (\(5 \times 6\)). This misinterpretation would lead to setting up completely wrong inequalities from the start.

Faltering Point 2: Misunderstanding the constraint "between 15 and 200"

Students might interpret "between" as exclusive (\(15 < \text{product} < 200\)) rather than inclusive (\(15 \leq \text{product} \leq 200\)). This would lead them to incorrectly exclude boundary cases where the product equals exactly 15 or 200, potentially eliminating valid solutions or including invalid ones.

Faltering Point 3: Setting up inequalities for individual bounds instead of the range

Students may try to find integers where \(\mathrm{n}\) is between certain values, rather than understanding that they need to find the smallest \(\mathrm{n}\) where \(\mathrm{n(n+1)} \geq 15\) AND the largest \(\mathrm{n}\) where \(\mathrm{n(n+1)} \leq 200\). They might incorrectly think they need to solve something like \(15 \leq \mathrm{n} \leq 200\) instead of the correct constraint on the products.

Errors while executing the approach

Faltering Point 1: Calculation errors in multiplication

Students may make basic arithmetic mistakes when computing products like \(13 \times 14\) or \(14 \times 15\). For instance, they might calculate \(13 \times 14 = 172\) instead of 182, or \(14 \times 15 = 200\) instead of 210. These errors would lead to incorrect boundary determinations.

Faltering Point 2: Testing insufficient boundary values

Students might test only one or two values around the estimated range rather than systematically checking both sides of each boundary. For example, they might check that \(\mathrm{n} = 4\) works (\(4 \times 5 = 20 \geq 15\)) but forget to verify that \(\mathrm{n} = 3\) doesn't work (\(3 \times 4 = 12 < 15\)), leading to incorrect minimum values.

Faltering Point 3: Incorrect estimation for the upper bound

When estimating where to start testing for the maximum value, students might incorrectly approximate \(\sqrt{200}\). They could estimate it as 10 or 15 instead of around 14, causing them to test values that are too far from the actual boundary and potentially miss the correct maximum value.

Errors while selecting the answer

Faltering Point 1: Confusing minimum and maximum in answer choices

Students might correctly calculate that the bounds are 4 and 13, but when looking at answer choices, accidentally select an option where these numbers are reversed or mixed up with their calculations. For example, they might choose an answer that has 13 as the minimum and 4 as the maximum.

Faltering Point 2: Selecting based on partial verification

Students might find that their calculated bounds work (\(4 \times 5 = 20\) and \(13 \times 14 = 182\) are both in range) but fail to verify that the adjacent values don't work. They might then second-guess themselves and select a "safer" option like 3 and 15, thinking it provides more buffer room around their constraints.