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please help

by grandh01 » Thu Jul 12, 2012 11:21 pm
If n and y are positive integers, and 450y=n^3, which of the following must be an integer?

1) y/3*2^2*5
2) y/3^2*2*5
3) y/3*2*5^2

a)none
b) I only
c) II only
d) III only
e) I, II, III

Thanks in advance
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by Anurag@Gurome » Thu Jul 12, 2012 11:32 pm
grandh01 wrote:If n and y are positive integers, and 450y=n^3, which of the following must be an integer?

1) (y/3)*(2^2)*5
2) [(y/3)^2]*2*5
3) (y/3)*2*(5^2)
I'm assuming the brackets in the options to remove ambiguities.
Now note that, whether the options will be integer or not depends on whether y/3 is integer or not, i.e. whether y is a multiple of 3 or not.

Now, 450 = 2*(3²)*(5²)
As 450y = cube of a positive integer, y must contain at least two 2s, one 3, and one 5.

Hence, y is always a multiple of 3.
Therefore, all the three can be integer.

The correct answer is E.
Anurag Mairal, Ph.D., MBA
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