2001 AMC 8 Problems/Problem 18

Problem

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5?

Solution 1

This is equivalent to asking for the probability that at least one of the numbers is a multiple of , since if one of the numbers is a multiple of , then the product with it and another integer is also a multiple of , and if a number is a multiple of , then since is prime, one of the factors must also have a factor of , and is the only multiple of on a die, so one of the numbers rolled must be a . To find the probability of rolling at least one , we can find the probability of not rolling a and subtract that from , since you either roll a or not roll a . The probability of not rolling a on either dice is . Therefore, the probability of rolling at least one five, and thus rolling two numbers whose product is a multiple of , is

Solution 2

The only way to get a multiple of 5 is to have at least one 5. If the first dice rolls a 5, there are 6 ways to get a multiple of 5. If the second dice rolls a 5, there are also 6 ways. However, there is one case that is repeated: both dice roll a 5. Therefore, there are 6 + 6 - 1 = 11, and there is a total of 6 x 6 ways, so the probability is \frac{11}{36}, \boxed{\text{D}} $

Video Solution

https://youtu.be/4aX9-DZHgNw Soo, DRMS, NM

See Also

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