Problem Solving

There are a few ways to break down the cases here, some a bit faster than what I'll do below, but you can essentially just enumerate all of the possibilities to get an answer. We have 10 choices for the first card, and 10 for the second, so we have 100 choices in total - that's our denominator. Now, to get a product which is a multiple of 6, one of our numbers needs to be a multiple of 2, and one (possibly the same number) needs to be a multiple of 3. So, considering what we might pick with our first card:

* if we pick an odd prime (11, 13, 17, 19), we'll need to pick a multiple of 6 with the second card, so we need to pick 12 or 18, for a total of 8 possibilities (just multiplying the number of choices we have for each card)
* if we pick 12 or 18, we can pick anything with the second card, for a total of 20 possibilities
* if we pick an even number not divisible by 3 (i.e. 14, 16, or 20), we need to pick a multiple of 3 with the second card (12, 15 or 18) for a total of 9 possibilities
* if we pick 15, we need to pick one of the five even numbers with the second card, for a total of 5 possibilities

and adding the results from each case, there are 42 ways to pick numbers that give a product divisible by 6, so the answer is 42/100.