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Five integers between 10 and 99, inclusive, are to be formed by using each of the ten digits exactly once in such a way that the sum of the five integers is as small as possible. What is the greatest possible integer that could be among these five numbers?
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:
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
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:
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.
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:
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
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:
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.
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:
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.
The greatest possible integer that could be among the five numbers is 59.
Looking at our answer choices:
Our answer matches choice C. 59.
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.
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.
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.
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).
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.
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.