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# "Colored chips": I used 16000 = (2 * blue chips) (5 * green) (4 * red). That gives either 4 or 1 for the red chips.

https://gre.magoosh.com/questions/297

# Question

According to the problem I have 16000 = (2*blue) (5*green) (4*red), which is the same as 16000 = (x*2) (x*5) (y*4). I can rewrite that as 400 = x^2 * y. So 400 can be written as (10 * 10 * 4) or (20 * 20 * 1). So the number of red chips could be either 4 or 1. Right?

Unfortunately, no.

The question states that "the product of the point values of the chips is 16,000." So we have to find the product of the point value of each chip individually—not the sum.

Your equation is the equivalent of summing the point values within a given color and then multiplying, not multiplying the individual chip values. If there are x blue chips, that means 4^x, not 4x.

For example, say there were three blue chips. They are worth 4 points. We want the product of these chips, so 4 * 4 * 4 = 4^3 = 64. Again, not adding three 4's together or 4 * 3 = 12.

Thus, the expression:

(4^x)(5^x)(2^y) = 16000

or

(20^x)(2^y) = 16000.

But that's a very complicated equation with which to wrestle. I actually think it's a mistake to introduce variables into this problem at all. In general, in prime factorizations problems, introducing variables is not helpful.

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