2024 AMC 8 Problem 5

Question

Aaliyah rolls two standard

6

-sided dice. She notices that the product of the two numbers rolled is a multiple of

6.

Which of the following integers cannot be the sum of the two numbers?

5

6

7

8

9

A multiple of

6

needs a factor of

2

and a factor of

3

List the possible sums from dice products divisible by

6

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For the product to be a multiple of

6

, the two dice together must supply a factor of

2

and a factor of

3

. The possible sums among the answer choices can occur as follows:

\begin{gathered} 5=2+3,\quad 7=1+6, \\ 8=2+6,\quad 9=3+6. \end{gathered}

There is no way to get sum

6

while also having a product divisible by

6

: the pairs with sum

6

are

(1,5),

(2,4),

(3,3),

(4,2),

and

(5,1)

, and none have product divisible by

6

.

Thus, B is the correct answer.

Problem 5 in other years