Re: For all positive integers x and y, the expression xy is defined as

Understanding the Definition

The expression xy represents the smallest multiple of y that is greater than or equal to x.

Applying the Definition to the Problem

We are given that 20k = 30. This means:

30 is a multiple of k.
30 is the least multiple of k that is greater than or equal to 20.
Finding Possible Values of k

Since 30 is a multiple of k, k must be a factor (divisor) of 30. The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.

Now we need to check which of these factors satisfy the condition that 30 is the least multiple of k that is greater than or equal to 20.

k = 1: 201 = 20 (not 30)
k = 2: 202 = 20 (not 30)
k = 3: 203 = 21 (not 30)
k = 5: 205 = 20 (not 30)
k = 6: 206 = 24 (not 30)
k = 10: 2010 = 20 (not 30)
k = 15: 2015 = 30 (This works! The least multiple of 15 greater than or equal to 20 is 30)
k = 30: 2030 = 30 (This works! The least multiple of 30 greater than or equal to 20 is 30)
If we go further, we see that if we want 30 to be a multiple of k, and be greater than or equal to 20, then k must be a factor of 30 that is greater than 20/2=10.
Since k must be a factor of 30, and greater than 10, the factors that work are 15 and 30.

Answer

There are two different positive integers k (15 and 30) for which 20k = 30.

Therefore, the correct answer is (B).