We are given that the mean of 5 positive integers is 20, and we need to determine the largest possible value of one of these integers.

Step 1: Sum of the numbers

Since the mean of the five integers is 20, the sum of the five integers is: $\text{Sum} = 5 \times 20 = 100$

Step 2: Denote the integers

Let the five integers be $x_1, x_2, x_3, x_4, x_5$, arranged in increasing order such that: $x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$ The sum of these integers is: $x_1 + x_2 + x_3 + x_4 + x_5 = 100$

Step 3: Maximizing $x_5$

To maximize $x_5$, we need to minimize the other integers $x_1, x_2, x_3, x_4$. The smallest possible values for these integers are 1, since the numbers must be positive integers. Hence, let: $x_1 = x_2 = x_3 = x_4 = 1$

Step 4: Solve for $x_5$

Substitute $x_1 = x_2 = x_3 = x_4 = 1$ into the sum equation: $1 + 1 + 1 + 1 + x_5 = 100$ Simplifying: $4 + x_5 = 100$ $x_5 = 100 - 4 = 96$

Thus, the largest possible value of one of the integers is $\boxed{96}$.

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Would you like to explore more about this problem, or do you have another question? Here are some related questions to consider:

1. How would the solution change if we had 6 numbers instead of 5?
2. What if the integers could be non-distinct (not necessarily in increasing order)?
3. How can we generalize this problem for $n$ integers with a given mean?
4. What is the smallest possible value of the largest number if we change the mean?
5. How does the sum of the numbers relate to the possible values of individual numbers?

Tip: When maximizing or minimizing one number in a set, try to minimize the others as much as possible, within the given constraints!