There are two ways xy can be a multiple of 4. First, if either x or y (or both) is a mulitple of 4, it doesn't matter what the other number is: a multiple of 4 times anything is a mulitple of 4. Second, if neither of the numbers are multiples of 4, but both are even (for instance, 2 and 22), the product will be a multiple of 4. We need to find the probability of each of those possibilities.

To find the probability that either x, y, or both is a multiple of 4, it's easiest to find the probability that NEITHER are multiples of 4. The probability that x is NOT a multiple of 4 is 3/4 (1/4 of numbers are multiples of 4), and the probability that y is NOT a multiple of 4 is also 3/4. Thus, the probability that NEITHER is a multiple of 4 is (3/4)(3/4) = 9/16. Thus, the probability that one or both of the numbers is a multiple of 4 is 1 - 9/16 = 7/16.

That leaves us to solve for the other possibility: that both numbers are even but not multiples of 4. In any sequence of 4 consecutive integers, one of the 4 will be an even number that is not a multiple of 4. Thus, 1/4 of the numbers between 1 and 20 (or 21 and 40) is an even non-multiple of 4. The probability that BOTH numbers have these characteristics is (1/4)(1/4) = 1/16.

The probability that the product is a multiple of 4, then, is the sum of our two probabilities:
7/16 + 1/16 = 8/16 = 1/2, choice (E).