Manhattan GMAT Challenge Problem of the Week – 09 Mar 2011

by Manhattan Prep, Mar 9, 2011

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Question

Each bead in an urn is marked with a distinct positive integer and colored according to that integers remainder after division by 5, as shown in the following table:

Remainder	Color
0	Red
1	Blue
2	Green
3	Yellow
4	Orange

Four blue beads, three green beads, two yellow beads, and one orange bead are withdrawn. If the product of the numbers on these beads is displayed on another bead, according to the rules above, then the color of that bead is...

(A) red

(B) blue

(C) green

(D) yellow

(E) orange

Answer

The key to this problem is to translate the game into more familiar mathematical language. First of all, each beads color corresponds to a remainder after division by 5. For instance, Red = R0 (remainder of 0), which means that the number in question is a multiple of 5. Blue = R1, Green = R2, etc.

We withdraw four blues, three greens, two yellows, and an orange. In other words, we have four R1 numbers, three R2 numbers, two R3 numbers, and one R4 number.

If we want to, we can pick actual numbers. It would be best to pick small numbersfor instance, R1 could actually be 1, because when you divide 1 by 5, you get a quotient of 0 and a remainder of 1. Likewise, R2 could be 2, R3 could be 3, and R4 could be 4.

Multiplying these numbers together, we get 1x1x1x1x2x2x2x3x3x4 = 8x9x4 = 72x4 = 288. The remainder after division by 5 would be 3, and the color of the bead would be yellow. We would get the same result without picking numbers, of coursewe would have to multiply the remainders together, which would give us R288, and then wed reduce that to R3. Either way, we have a yellow bead.

The correct answer is D.

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