## Online Course Discussion Forum

### AMC12B FALL 2021 #11

Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by

I am really confused how things work here.

I tried to do complementary counting. There are a total of 6^6 ways, which is the denominator. And for the numerator, one of the dice can be 1,2,3,5,or 6, and the rest have to be odd, so it's 5*3^5 ways. Then I tried to do (6^6-5*3^5)/6^6, which is 187/192.

It was not the correct answer, which is 59/64.

The idea of complementary counting is correct.

To count the cases that the product is NOT divisible by $4$, there can be at most one factor of $2$. So far so good.

However, it is not as simple as you stated. One of the dice can be 1,2,3,5, or 6---this is true, but you also need to choose that die, so you need to multiply by $6$. That is not all the problems here. When this die has number 1, 3, or 5, it is odd, same as the other dice, so there is a lot of overcounting here.

To count it in a clean way, consider the cases (1) All dice show odd numbers, and (2) One of the dice is 2 or 6, and all others are odd. Then you can subtract both cases from the total.

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