Five integers between 10 and 99, inclusive, are to be formed by using each of the ten digits exactly once...

1. Translate the problem requirements: We need to form five 2-digit numbers using each digit 0-9 exactly once, arrange them so their sum is minimized, and find the largest possible number among these five.
2. Establish the sum minimization strategy: To minimize the total sum, place the smallest digits (0,1,2,3,4) in the tens places and larger digits (5,6,7,8,9) in the units places.
3. Handle the zero constraint: Since 0 cannot be a tens digit, use digits 1,2,3,4 for tens places, putting 0 in a units place and promoting the next smallest digit (5) to a tens place.
4. Maximize one number within the constraint: To find the greatest possible number, systematically test placing larger tens digits (4, then 5) and pairing them with the largest available units digit while maintaining minimum sum.
5. Verify the optimal configuration: Confirm that our arrangement truly minimizes the sum and that no larger number is possible under this constraint.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to do in everyday terms. We have the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and we need to use each digit exactly once to create five two-digit numbers. Think of it like having ten number cards that we need to arrange into five pairs.

Our goal has two parts:

• Make the sum of these five numbers as small as possible
• Among these five numbers that give us the minimum sum, find the largest possible individual number

For example, if we could form numbers like 10, 23, 45, 67, 89, their sum would be 234. But we want to find the arrangement that gives us the smallest possible sum, and then see what's the biggest individual number we could have in that optimal arrangement.

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

2. Establish the sum minimization strategy

To minimize the total sum, we need to think about place value. In a two-digit number, the tens digit contributes 10 times more to the value than the units digit. For example, in the number 47, the 4 contributes 40 to the total while the 7 contributes only 7.

This means to minimize our sum, we should put the smallest available digits in the tens places and the larger digits in the units places.

Ideally, we'd want:

• Tens places: 0, 1, 2, 3, 4 (the five smallest digits)
• Units places: 5, 6, 7, 8, 9 (the five largest digits)

This would give us numbers like: 05, 16, 27, 38, 49
But wait - we can't have 05 as a two-digit number because that's really just 5, which is a one-digit number.

3. Handle the zero constraint

Since 0 cannot be in the tens place (it would make a one-digit number), we need to adjust our strategy.

We'll use 1, 2, 3, 4 in the tens places, and we need one more tens digit. The next smallest available digit is 5, so 5 must also go in a tens place.

This means:

• Tens places: \(\{1, 2, 3, 4, 5\}\)
• Units places: \(\{0, 6, 7, 8, 9\}\)

So our five numbers would be: 10, 26, 37, 48, 59 (in some arrangement of the units digits)
The sum would be: \(10 + 20 + 30 + 40 + 50 + 0 + 6 + 7 + 8 + 9 = 180\)

Process Skill: APPLY CONSTRAINTS - Recognizing that 0 cannot be a leading digit

4. Maximize one number within the constraint

Now we need to find the largest possible individual number while maintaining this minimum sum strategy.

Let's think systematically. To get the largest possible number, we want the largest possible tens digit paired with the largest possible units digit.

From our required tens digits \(\{1, 2, 3, 4, 5\}\), the largest is 5.
From our required units digits \(\{0, 6, 7, 8, 9\}\), the largest is 9.

So we can form the number 59.

Let's verify this works by showing a complete valid arrangement:

• 10 (using tens digit 1, units digit 0)
• 26 (using tens digit 2, units digit 6)
• 37 (using tens digit 3, units digit 7)
• 48 (using tens digit 4, units digit 8)
• 59 (using tens digit 5, units digit 9)

Sum: \(10 + 26 + 37 + 48 + 59 = 180\)

Can we do better than 59? Could we get a number in the 60s, 70s, 80s, or 90s?

For a number in the 60s or higher, we'd need 6, 7, 8, or 9 in a tens place. But if we put any of these larger digits in a tens place instead of 5, we'd increase our sum beyond the minimum, which violates our constraint.

5. Verify the optimal configuration

Let's double-check our reasoning:

1) Our sum of 180 is indeed minimal because we used the five smallest possible digits (1,2,3,4,5) in the tens places.

2) Within this constraint, 59 is the largest possible number because:

• 5 is the largest tens digit we can use (since 6,7,8,9 must be in units places)
• 9 is the largest units digit available

3) Any attempt to create a larger individual number (like 69, 78, 89, 90s) would require moving a digit from the units place to the tens place, increasing our total sum above the minimum.

Therefore, 59 is our answer.

Final Answer

The greatest possible integer that could be among the five numbers is 59.

Looking at our answer choices:

1. 98
2. 91
3. 59
4. 50
5. 37

Our answer matches choice C. 59.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the optimization problem structure

Students often focus only on finding the minimum sum without recognizing this is a two-step optimization problem. They may find arrangements that minimize the sum but fail to understand they need to then maximize one individual number within that constraint. This leads them to stop after finding any valid minimum-sum arrangement without exploring which arrangement gives the largest possible individual number.

2. Incorrectly handling the zero constraint

Many students initially think they can use digits 0,1,2,3,4 in the tens places to minimize the sum, forgetting that 0 cannot be a leading digit in a two-digit number. This fundamental oversight leads them down the wrong path, potentially concluding they can achieve a smaller sum than actually possible, which then affects their entire solution strategy.

3. Misinterpreting what "greatest possible integer" means in context

Some students may think they need to find the greatest integer across ALL possible arrangements of the digits, rather than understanding it means the greatest integer specifically among arrangements that achieve the minimum sum. This misinterpretation leads them to focus on creating numbers like 98 or 91 without considering the sum minimization constraint.

Errors while executing the approach

1. Incorrectly assigning digits to tens and units places

Even when students understand they need \(\{1,2,3,4,5\}\) in tens places and \(\{0,6,7,8,9\}\) in units places, they may make errors in the specific pairing. For example, they might pair 5 with a smaller units digit like 6 or 7, failing to recognize that to maximize one number, they need to pair the largest available tens digit (5) with the largest available units digit (9).

2. Arithmetic errors in sum verification

Students may make simple addition mistakes when calculating the total sum to verify their arrangement achieves the minimum. Since the sum verification is crucial for confirming the approach is correct, arithmetic errors here can lead to doubt about whether their digit assignment strategy is optimal.

Errors while selecting the answer

1. Choosing a number from a non-optimal arrangement

Students may correctly identify that arrangements like 37 or 48 are possible in a minimum-sum configuration, but fail to recognize that 59 is also achievable and is larger. They might select answer choice E (37) or D (50) because these seem reasonable without fully exploring all possible pairings within the optimal tens/units digit assignment.