Walkabout wrote:
In a certain game, a large container is filled with red, yellow, green, and blue beads worth, respectively, 7, 5, 3, and 2 points each. A number of beads are then removed from the container. If the product of the point values of the removed beads is 147,000, how many red beads were removed?
(A) 5
(B) 4
(C) 3
(D) 2
(E) 0
The first thing we want to do is to decipher the problem. The key word here is “product;" we are told that the product of the point values of the removed beads is 147,000.
Let's look at an example. Let’s say 2 red beads, 2 yellow beads, 2 green beads, and 2 blue beads were removed. The product of the point values would be:
7 x 7 x 5 x 5 x 3 x 3 x 2 x 2 = 44,100
Conversely, if we were to take 44,100 and break it down into its prime factors, we would get:
44,100 = 7 x 7 x 5 x 5 x 3 x 3 x 2 x 2
Note that this prime factorization of 44,100 corresponds exactly to the point values of the 2 red beads, the 2 yellow beads, the 2 green beads, and the 2 blue beads from the example.
This example is important because we can now use the same approach for answering the actual question. We are given that the product of the point values of the removed beads is 147,000. Thus, some number of 7’s, 5’s, 3’s, and 2’s is multiplied together to equal 147,000. Since we only care about the red beads, we only care about the number of 7’s in the product. Let’s keep this in mind as we break down 147,000.
147,000 = 147 x 1,000 = 49 x 3 x 1,000 = 7 x 7 x 3 x 1,000
Since 1,000 does not have 7 as a factor, we see that there are two 7’s in the product of 147,000; thus, 2 red beads were removed.
The answer is D.