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# The positive integers x,y, and z are such that x is a factor

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Senior Manager
Joined: 05 Jun 2005
Posts: 455
Followers: 1

Kudos [?]: 22 [0], given: 0

The positive integers x,y, and z are such that x is a factor [#permalink]  01 Nov 2006, 21:37
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Question Stats:

100% (02:48) correct 0% (00:00) wrong based on 2 sessions
The positive integers x,y, and z are such that x is a factor of y and y is a factor of z. Is z even?

1) xz is even
2) y is even.
Manager
Joined: 17 Dec 2004
Posts: 73
Followers: 2

Kudos [?]: 6 [0], given: 0

I get D.

To start, I applied the following rules:

Even x Even = Even
Even x Odd = Even
Odd x Odd = Odd

With that in mind...

Statement 1:

xz is even. This tells us that x and z are both even or one is even, and the other is odd.

From the question, we know that x and y are factors of z, and as such, the lowest value for z is xy. Replacing z with xy, we find that unless both x and y are odd, xy (thus z) must be even. Because statement 1 tells us that xz is even, x and y (the factors of z) can't both be odd. Going back to Even x Odd = Even, xy must be even. As a result, z must be even. Sufficient.

Statement 2:

Y is even. Going back to the rules above, we know that anything multiplied by an even number is even. Since y is a factor of z, it follows that z must be even. Sufficient.

Since both are sufficient on their own, I take answer D.
Director
Joined: 01 Oct 2006
Posts: 500
Followers: 1

Kudos [?]: 18 [0], given: 0

St xz is even possible only one is even or both are even or odd even.
When x is even y is even and thus z is even when x is odd z has to be even.
thus suff

St2 states y is even thus x and z have to even

Director
Joined: 06 Sep 2006
Posts: 745
Followers: 1

Kudos [?]: 17 [0], given: 0

1) Since x is a factor of z. All the factors of an odd number are always odd. Since x is a factor of z and if z is odd x has to be odd but
that can not be true because that will make 'xz' odd. So they are both even. SUFFICIENT.

2) Odd numbers can not have even factors. So, z is even.

D
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