To determine the greatest number of items that could have been placed on each plate, we need to find the greatest common divisor (GCD) of the number of each type of donated item. This is because the GCD will tell us the largest number that evenly divides each of the quantities, ensuring there are no leftovers.

Let's denote the number of each type of donated item as $a_1, a_2, \ldots, a_n$.

The problem does not specify the actual numbers of each type of item, so let's consider a generic case with three types of items: $a, b,$ and $c$.

To find the GCD of these three numbers:

1. Calculate the GCD of $a$ and $b$, which we'll call $d_1 = \text{GCD}(a, b)$.
2. Calculate the GCD of $d_1$ and $c$, which we'll call $d = \text{GCD}(d_1, c)$.

The value $d$ is the greatest number of items that could have been placed on each plate.

For example, if the donated items are:

• 48 cookies
• 64 brownies
• 80 cupcakes

We proceed as follows:

1. Find the GCD of 48 and 64: $\text{GCD}(48, 64) = 16$

2. Find the GCD of the result and 80: $\text{GCD}(16, 80) = 16$

So, the greatest number of items that could have been placed on each plate is 16.

In general, without specific numbers for $a, b, \text{and } c$, the solution requires calculating the GCD of all given quantities of donated items.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. How do you find the GCD of more than two numbers?
2. Can the GCD be larger than the smallest number in a set of numbers?
3. How does the GCD of two numbers relate to their prime factorizations?
4. What if the number of each type of item is a prime number?
5. How would you adapt the solution if there were leftovers?

Tip: Understanding how to calculate the GCD using the Euclidean algorithm can simplify finding the GCD of large numbers.