Problem:
Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
We will use complementary counting to find the probability that the product is not divisible by . Then, we can find the probability that we want by subtracting this from . We split this into cases:
The product is not divisible by .
We need every number to be odd, and since the chance we roll an odd number is , our probability is
The product is divisible by , but not by .
We need numbers to be odd, and one to be divisible by , but not by . There is a chance that an odd number is rolled, a chance that we roll a number satisfying the second condition (only 2 and 6 work), and 6 ways to choose the position in which the even number appears.
Our probability is:
Therefore, the probability that the product is not divisible by is:
Our answer is:
The problems on this page are the property of the MAA's American Mathematics Competitions