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Factors Vs Prime Factors

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Factors Vs Prime Factors

by Leon1984 » Wed Oct 28, 2009 4:47 am
If x and y are positive integers, is x/y an integer?
(i) Every factor of y is also a factor of x
(ii) Every prime factor of y is also a prime factor of x

Please explain why (ii) is insufficient while (i) is?
My thoughts were B or D, but it turned out that the answer is A. Please explain.

Thank you
Leon
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by mp2437 » Wed Oct 28, 2009 6:07 am
(i)

If 2 numbers have every factor in common, then the ratio of the number with more total factors to a number with less factors will result in a number greater than or equal to 1. Choose a few examples to see this, say y = 15 (factors of 1,3,5,15), and x = 30 (factors of 1,2,3,5,6,10,15,30). You can see that x/y = 30/15 = 2 (an integer), and it satisfies statement 1 which says every factor of y is a factor of x. if x = 15, it would satisfy the equation as well, and will give you an integer (x/y = 15/15 = 1).

(ii)

it is not true that an integer will be formed by the ratio of 2 numbers if only their prime factors are common. For example, say y = 8 (prime factor of 2), and x = 30 (prime factors of 2,3,5). Statement 2 is satisfied such that every prime factor of y (2) is also a prime factor of x (x = 30 does in fact have a prime factor of 2), but x/y = 30/8 is not an integer, so statement (ii) is false.

Hope this helps.
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by gnet » Wed Oct 28, 2009 6:33 am
Leon1984 wrote: ii) Every prime factor of y is also a prime factor of x
mp2437 wrote: (ii)it is not true that an integer will be formed by the ratio of 2 numbers if only their prime factors are common. For example, say y = 8 (prime factor of 2), and x = 30 (prime factors of 2,3,5). Statement 2 is satisfied such that every prime factor of y (2) is also a prime factor of x (x = 30 does in fact have a prime factor of 2), but x/y = 30/8 is not an integer, so statement (ii) is false.
I think (ii) can be eliminated only if it means -- Every distinct prime factor of y is also a distinct prime factor of x. For example, 4 has total two prime factors (2x2), but a single distinct prime factor (2).

In the example given by mp2437, if we were to consider
  • Prime factors of (x) 30 = 2 x 3 x 5
    Prime factors of (y) 8 = 2 x 2 x 2
..then every prime factor of y is not a prime factor of x, and so our example is invalid to determine sufficiency.

However if we consider
  • Every distinct prime factor of 30 = 2,3,5
    Every distinct prime factor of 8 = 2
.. then we have an example that proves x/y (i.e. 30/8) not to be an integer.
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by mp2437 » Wed Oct 28, 2009 6:57 am
gnet, that's a good catch. I thought of it in terms of distinct prime factors, and if the OA stands as A, it looks like the question also assumes that it is in fact distinct prime factors that matter.
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by gnet » Wed Oct 28, 2009 7:22 am
mp2437 wrote:gnet, that's a good catch. I thought of it in terms of distinct prime factors, and if the OA stands as A, it looks like the question also assumes that it is in fact distinct prime factors that matter.
thanks mp2437. I knew you were talking about distinct prime factors only :) I think the actual GMAT would have the answer choice worded better to make it explicitly clear.
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by Leon1984 » Wed Oct 28, 2009 7:52 am
Thank you guys. I think that the word "every" confused me, for some reason I assumed that when they say every prime factor they mean that it also has the same number of times this factor appears.

In the example which you have provide, if y=8 which is 2*2*2, x will also have 2*2*2 in it, since they said every prime factor.
Leon
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