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siddhantvarma
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Five integers between 10 and 99, inclusive, are to be formed by using 07 Aug 2024, 23:35
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I get the logic here that to minimize the sum of the five integers, we will make the tens digits as smallest possible which means we'll use 1, 2, 3, 4, and 5 as the TENS digits and use the other five possible digits (0, 6, 7, 8, and 9) for the UNITS digits. However, how do we know that the sum of all possible combinations of 5 two-digit numbers will be the same, ie 180? While attempting this question, I marked 50 because I did not calculate if other combinations where the largest two-digit number is 59 yield the same sum, ie 180. My question then becomes at what point do we decide that let me try to sum up other combinations as well to see if they give the same sum or a larger sum? I assumed that the combination with 50 the largest number, the sum will be minimum. Is it just a miss on my end of not checking all the cases (doing this in time constraints), or it is something we intuitively derive by looking at the possible combinations that it might be worth checking if all combinations give the same sum?­
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Five integers between 10 and 99, inclusive, are to be formed by using 15 Jan 2025, 06:15
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gmatt1476
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?

A. 98
B. 91
C. 59
D. 50
E. 37

PS30402.01
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59+40+38+27+16=180 and 19+28+37+46+50 =180 also so I think answers can be c and d
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Five integers between 10 and 99, inclusive, are to be formed by using 15 Jan 2025, 14:17
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We want the greatest possible integer among 5 numbers, case 1 gives 59 and case 2 gives 50; 59 > 50, hence we consider only the case that has 59. Answer C.
cian12
gmatt1476
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?

A. 98
B. 91
C. 59
D. 50
E. 37

PS30402.01
59+40+38+27+16=180 and 19+28+37+46+50 =180 also so I think answers can be c and d
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Five integers between 10 and 99, inclusive, are to be formed by using 15 Jan 2025, 14:25
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You don't need to check all combinations - when you add all the units digits (0, 6, 7, 8, and 9) you will get 30, and 3 will be carried over to be added with tens digits (1, 2, 3, 4, and 5) which will give you 18. Irrespective of the combination of these 5 two digit numbers, that's how you know that sum will always be 180. Now, you want the largest digit, so combine the largest unit digit and largest tens digit => 59
siddhantvarma
I get the logic here that to minimize the sum of the five integers, we will make the tens digits as smallest possible which means we'll use 1, 2, 3, 4, and 5 as the TENS digits and use the other five possible digits (0, 6, 7, 8, and 9) for the UNITS digits. However, how do we know that the sum of all possible combinations of 5 two-digit numbers will be the same, ie 180? While attempting this question, I marked 50 because I did not calculate if other combinations where the largest two-digit number is 59 yield the same sum, ie 180. My question then becomes at what point do we decide that let me try to sum up other combinations as well to see if they give the same sum or a larger sum? I assumed that the combination with 50 the largest number, the sum will be minimum. Is it just a miss on my end of not checking all the cases (doing this in time constraints), or it is something we intuitively derive by looking at the possible combinations that it might be worth checking if all combinations give the same sum?­
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Five integers between 10 and 99, inclusive, are to be formed by using 22 Sep 2025, 23:10
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Hey there! This is a classic optimization problem that often trips students up because it's actually asking you to solve two different objectives at once - and many test-takers miss this crucial detail.

Let's think about this step by step to see what's really happening here:

Step 1: Understand What We're Really Optimizing

You need to form five 2-digit numbers using each digit 0-9 exactly once, but notice the question has a double requirement:
- First: Make the sum as small as possible
- Second: Among all arrangements that achieve this minimum sum, find the largest individual number

This isn't just "find the minimum sum" - it's "find the minimum sum, THEN maximize one number within that constraint."

Step 2: Strategy for Minimum Sum

To minimize the total sum, you want the smallest digits in the tens places (since tens digits contribute 10x more to the value). Ideally, you'd use digits {0,1,2,3,4} for tens places and {5,6,7,8,9} for units places.

But here's the catch - you can't have 0 as a tens digit (that would make a 1-digit number, not a 2-digit number).

So you're forced to use: {1,2,3,4,5} in tens places and {0,6,7,8,9} in units places.

Step 3: Maximize Within the Constraint

Now, to get the largest possible individual number while maintaining this minimum sum strategy:
- Largest available tens digit: 5
- Largest available units digit: 9
- Therefore, the largest possible number is 59

A complete valid arrangement would be: 10, 26, 37, 48, 59
Sum = \(10 + 26 + 37 + 48 + 59 = 180\)

Step 4: Verify This is Optimal

Could we get a number in the 60s, 70s, 80s, or 90s? No - that would require putting 6, 7, 8, or 9 in a tens place, which would increase our sum beyond the minimum.

Answer: C (59)

The key insight here is recognizing this as a constrained optimization problem where you need to work within the minimum-sum constraint to find your answer.

You can check out the complete framework and systematic approach on Neuron by e-GMAT to master constrained optimization problems and see how this pattern applies to similar questions. You can also practice with comprehensive solutions for many other official questions here to build consistent accuracy on these multi-step optimization problems.
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Five integers between 10 and 99, inclusive, are to be formed by using 19 Aug 2026, 08:57
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In this question, why cant the five integers be 10,20,30,40 & 50 and the highest be 50. As far as i read the question they are only asking not to repeat the tens digit. Nothing is mentioned in the unit digit.

gmatt1476
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?

A. 98
B. 91
C. 59
D. 50
E. 37

PS30402.01
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Five integers between 10 and 99, inclusive, are to be formed by using 19 Aug 2026, 09:01
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Bittoworkspace
In this question, why cant the five integers be 10,20,30,40 & 50 and the highest be 50. As far as i read the question they are only asking not to repeat the tens digit. Nothing is mentioned in the unit digit.


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Your doubt is addressed in the thread several times. Please review. Hope this helps.
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Five integers between 10 and 99, inclusive, are to be formed by using 19 Aug 2026, 09:19
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My confusion still remains after going through the thread. The question does not put the limitation to not repeating 0 in unit place. Also the question asks to minimize the sum and share the highest digit out of the 5 - 2 digits integers. In such a case why 10,20,30,40,50 be a 5 integers.
Bunuel


Your doubt is addressed in the thread several times. Please review. Hope this helps.
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Five integers between 10 and 99, inclusive, are to be formed by using 19 Aug 2026, 09:23
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Bittoworkspace
My confusion still remains after going through the thread. The question does not put the limitation to not repeating 0 in unit place. Also the question asks to minimize the sum and share the highest digit out of the 5 - 2 digits integers. In such a case why 10,20,30,40,50 be a 5 integers.

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You are not reading the question and the replies in the thread carefully enough.

The question says “each of the ten digits exactly once,” meaning the ten digits 0, 1, 2, ..., 9, not the tens digits.

So 10, 20, 30, 40, and 50 are not allowed because 0 is repeated five times, while 6, 7, 8, and 9 are not used at all.
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