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We have ten digits from 0 to 9 to form 5 two-digit integers so that each digit is used once
0 cannot be the tens digit
In order for the sum of 5 integers to be the smallest
=> The 5 tens digits must be 1, 2, 3, 4, 5
=> The 5 unit digits must be 0, 6, 7, 8, 9
We can combine any ten digit with any unit digit (just need to ensure each is used once) and still get the same sum
=> The greatest possible integer is 59
0 cannot be the tens digit
In order for the sum of 5 integers to be the smallest
=> The 5 tens digits must be 1, 2, 3, 4, 5
=> The 5 unit digits must be 0, 6, 7, 8, 9
We can combine any ten digit with any unit digit (just need to ensure each is used once) and still get the same sum
=> The greatest possible integer is 59
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?
