### 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 positive integers less than 100 that have a product that is a multiple of 3.
To do that, I need to think about the factors of 3 and how they can appear in the product.
If one of the factors is a multiple of 3, then the product will be a multiple of 3 regardless of the other factor.
If neither factor is a multiple of 3, then the product will not be a multiple of 3.
So, the only way the product can fail to be a multiple of 3 is if both factors are not multiples of 3.
How many multiples of 3 are there less than 100?
Well, the smallest one is 3 and the largest one is 99, so there are 33 of them.
How many non-multiples of 3 are there less than 100?
Well, there are 99 total positive integers less than 100, so there are 99 - 33 = 66 of them.
How many ways can I choose two non-multiples of 3 from 66 of them?
Well, that's the same as choosing two numbers from a set of 66, which is 66 choose 2, or 66 times 65 divided by 2, or 2145.
How many ways can I choose two numbers from 99 of them?
Well, that's the same as choosing two numbers from a set of 99, which is 99 choose 2, or 99 times 98 divided by 2, or 4851.
So, the probability that both factors are not multiples of 3 is 2145 divided by 4851, or about 0.4425.
Therefore, the probability that the product is a multiple of 3 is 1 minus 0.4425, or about 0.5575.