### Instruction: Two different positive integers less than 100 are randomly chosen and multiplied together. What is the probability that the resulting product is a multiple of 3?### Response: I need to find the fraction of pairs of integers that have a product that is a multiple of 3, out of all possible pairs of integers less than 100.
A product is a multiple of 3 if and only if at least one of the factors is a multiple of 3.
So I can count how many factors are multiples of 3 and how many are not, and then use the multiplication principle to find the number of pairs.
There are 33 multiples of 3 between 1 and 99, inclusive, because I can divide 99 by 3 and get 33.
There are 66 factors that are not multiples of 3, because I can subtract 33 from 99.
To form a pair that has a product that is a multiple of 3, I can either choose both factors to be multiples of 3, or one factor to be a multiple of 3 and the other not.
The number of ways to choose both factors to be multiples of 3 is 33 times 32, divided by 2 to avoid double counting.
This is 528.
The number of ways to choose one factor to be a multiple of 3 and the other not is 33 times 66, which is 2178.
So the total number of pairs that have a product that is a multiple of 3 is 528 plus 2178, which is 2706.
The total number of possible pairs of integers less than 100 is 99 times 98, divided by 2 to avoid double counting.
This is 4851.
Therefore, the probability that the product is a multiple of 3 is 2706 divided by 4851, which is approximately 0.5577.