2017 AMC 8
Complete problem set with solutions and individual problem pages
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?
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Solution 1
The green marbles and yellow marbles form of the total marbles. Now, suppose the total number of marbles is . We know the number of yellow marbles is and a positive integer. Therefore, must divide . Trying the smallest multiples of for , we see that when , we get there are yellow marbles, which is impossible. However when , there are yellow marbles, which must be the smallest possible.
Solution 2
Since of the marbles are blue and are red, it is clear that the total number of marbles must be divisible by . If there are marbles, then are blue, are red, and are green, meaning that there are yellow marbles. This is impossible. Trying the next multiple of , , we find that are green, are red, and are green, meaning that the minimum number of yellow marbles is .
