5. The sum of 7 unequal positive integers is 61. What is the largest possible value that any of those integers can have?

A.55
B.40
C.39
D.13
E.9
please show work

To find the largest possible value that any of the integers can have, we need to distribute the sum of 61 as evenly as possible among the 7 integers. Since the integers are unequal, we want to maximize the value of one of them while minimizing the others.

To maximize the value of one integer, we should minimize the values of the others. In this case, we want to distribute 61 as evenly as possible by assigning the smallest possible values to the remaining 6 integers.

To do this, we can start by assigning the smallest possible value to one of the integers, which is 1. This leaves us with 61 - 1 = 60 to distribute among the remaining 6 integers.

To distribute 60 as evenly as possible, we can start by assigning the second smallest possible value to the next integer, which is 2. This leaves us with 60 - 2 = 58 to distribute among the remaining 5 integers.

Continuing this pattern, we assign the third smallest possible value to the next integer, which is 3. This leaves us with 58 - 3 = 55 to distribute among the remaining 4 integers.

Continue this process until all integers have been assigned values.

1 + 2 + 3 + 4 + 5 + 6 + x = 61, where x is the value of the largest integer.

Since the sum of the first 6 integers is 21, we have:

21 + x = 61

Subtracting 21 from both sides:

x = 40

Therefore, the largest possible value that any of the integers can have is 40.

The correct answer is B.40.

61 - (1 + 2 + 3 + 4 + 5 + 6) = ?