If the sum of ten different positive integers is 100, what is the greatest possible value of any one of these ten integers?

Introduction / Context:
This is a classic optimization problem under constraints. We are given ten different positive integers with a fixed total sum, and we must maximize the largest of them. It tests the learner's ability to reason about how to distribute a fixed total among several distinct positive numbers in order to maximize one of them, which is a common theme in combinatorics and logical reasoning questions.

Given Data / Assumptions:

• There are exactly ten positive integers.
• All ten integers are distinct.
• Their total sum is 100.
• We need the greatest possible value of the largest integer among them.

Concept / Approach:
To maximize one number while keeping the total sum fixed and all numbers different and positive, we should minimize the other nine numbers subject to the constraints. The smallest possible set of nine distinct positive integers is 1, 2, 3, 4, 5, 6, 7, 8, and 9. Once we fix these nine integers, the tenth number is forced by the total sum. This gives the maximum possible value for the largest number, because any attempt to increase it further would require decreasing one of the others below these minimal values or repeating a value, which is not allowed.

Step-by-Step Solution:
Choose the nine smallest distinct positive integers: 1, 2, 3, 4, 5, 6, 7, 8, and 9.Compute their sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.This sum equals 45.Let the tenth number be M.The total sum of all ten integers is 100.So 45 + M = 100.Therefore, M = 100 - 45 = 55.Thus, the largest possible value of any one integer is 55.

Verification / Alternative check:
Check distinctness: the set of ten numbers is {1, 2, 3, 4, 5, 6, 7, 8, 9, 55}, which are all distinct positive integers.Check the sum: 45 + 55 = 100, which matches the required total.If we tried to make the largest number greater than 55, the sum of the remaining nine numbers would have to be less than 45.However, the minimal possible sum of nine distinct positive integers is already 45, so we cannot reduce that further, confirming that 55 is the maximum possible value.

Why Other Options Are Wrong:
91, 64, and 45 do not represent the true maximum.For example, if the largest number was 64, the remaining nine distinct positive integers would have to sum to 36, which is impossible because the smallest such sum is 45.Similarly, 91 would leave only 9 for nine different positive integers, which is clearly not feasible.

Common Pitfalls:
A common error is to forget that the integers must be distinct and to use the same small number multiple times, which can falsely suggest a larger maximum.Another mistake is to pick arbitrary small numbers instead of the nine smallest possible positive integers.Miscounting the sum of 1 through 9 can also lead to incorrect conclusions about the maximum value.

Final Answer:
The greatest possible value among the ten different positive integers is 55.