If x is a randomly chosen integer between 1 and 20, inclusive, and y is a randomly chosen integer between 21 and 40, inclusive, what is the probability that xy is a multiple of 4?
Select the correct option.
A) 1/4 B) 1/3 C) 3/8 D) 7/16 E) ½
There are two ways that I see of getting a multiple of 4   1) x and/or y is a multiple of 4   2) x is a multiple of 2 and y is a multiple of 2   for 1 we have   P(option 1) = (5/20)(20/20) + (5/20)(20/20) - (5/20)(5/20) = 1/2 - 1/16   since 5 of the numbers in each range are divisible by 4, and if either x or y is selected to be divisible by 4 the selection of the other variable does not matter. Notice the last subtraction term which corrects the fact that we have counted the possibility of both being divisible by 4 twice (e.g. P(A union B) = P(A) + P(B) - P(A and B)).   Notice at this point that the probability of option one is already 7/16, the second largest available option. If we can come up with any other possibilities for getting xy divisble by 4 we must go with option E.   Of course this is the case because if we select a multiple of 2 that is not a multiple of 4 from both ranges of numbers, the product is divisible by 4.   We can formally compute this as:   P(option 2) = (5/20) * (5/20) = 1/16   since we do not wish to double count possibilities that fall into category 1, we only count the probability of selecting even numbers from each set that are not divisible by 4 (e.g. the other 5 even integers in each set).   Total probability is then P(option 1) + P(option 2) = 1/2
<result>E</result>
If x is a randomly chosen integer between 1 and 20, inclusive, and y is a randomly chosen integer between 21 and 40, inclusive, what is the probability that xy is a multiple of 4?