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# Number properties - factors/multiples

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Intern
Joined: 28 Mar 2012
Posts: 2

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29 Mar 2012, 11:04
Is integer (x^2)*(y^4) divisible by 9?
 x is an integer divisible by 3
 xy is an integer divisible by 9

My thinking was that:

if 3 is a factor of x, than x^2 will have atleast 2 3's as its prime factors - so x^2 is divisible by 9, so the entire expression will be divisible by 9 - statement 1 is sufficient.

if xy is divisible by 9, then the expression can be rewritten as (xy)*(x*(y^3)) = 9m(x*(y^3)) which is divisible by 9 - statement 2 is sufficient.

But the answer is that both statements are not sufficient together..what am i doing wrong? fyi - this question is one of the gmatclub flashcards
SVP
Joined: 14 Apr 2009
Posts: 2085
Location: New York, NY
Re: Number properties - factors/multiples [#permalink]

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29 Mar 2012, 14:39
The main issue here to consider is whether the variables are integers.
With (1), x IS an integer
With (2) xy is an integer, but this does not mean X and Y are both integers. x can be 0.5 and y can be 2, we still have xy as an integer.

Does it make a difference?

Yes. 0.5^2 = some complex number with decimals. Is a decimal number divisible by 9? No. If we multiply that decimal number by 9, is it divisible by 9? Still no.
But if x and y are integers AND xy is an integer that is divisible by 9, then we have Yes as an answer.
Thus (2) gives us mixed answers - both yes and no - thus inconsistent. Not sufficient.

With (1), the same concept applies. x is an integer but y can be a fraction.
The moment you take a decimal and put a 4th exponent to it, it's one complex decimal. And it can't be divisible by 9. Thus yes it works in the case you may have calculated, but it doesn't work in the case of a decimal.

Thus, inconsistent.

E - since both choices give us inconsistent results.
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Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India
Re: Number properties - factors/multiples [#permalink]

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30 Mar 2012, 04:20
shreya32 wrote:
Is integer (x^2)*(y^4) divisible by 9?
 x is an integer divisible by 3
 xy is an integer divisible by 9

My thinking was that:

if 3 is a factor of x, than x^2 will have atleast 2 3's as its prime factors - so x^2 is divisible by 9, so the entire expression will be divisible by 9 - statement 1 is sufficient.

if xy is divisible by 9, then the expression can be rewritten as (xy)*(x*(y^3)) = 9m(x*(y^3)) which is divisible by 9 - statement 2 is sufficient.

But the answer is that both statements are not sufficient together..what am i doing wrong? fyi - this question is one of the gmatclub flashcards

Probably, the thing that confused you was that \(x^2*y^4\) is an integer (given in the question stem) and stmnt 1 says that x is an integer. One might feel that it implies that y is also an integer. This is the reason number properties questions are tricky.

Say, x = 81, y = 1/9
x is an integer divisible by 3.
\(x^2*y^4 = 81*81*(1/9)*(1/9)*(1/9)*(1/9) = 1\) is an integer.
But y is not an integer and \(x^2*y^4\) is not divisible by 9.

Using the same example, you can see why both the statements together are not sufficient.
x = 81, y = 1/9
x is an integer divisible by 3.
\(x^2*y^4 = 81*81*(1/9)*(1/9)*(1/9)*(1/9) = 1\) is an integer.
xy = 81*(1/9) = 9 is an integer divisible by 9.
But \(x^2*y^4\) is not divisible by 9.
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Re: Number properties - factors/multiples   [#permalink] 30 Mar 2012, 04:20
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