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by theachiever » Tue Dec 18, 2012 1:54 am
If Z is an integer, is 22 a factor of Z?

(1) 22 is a factor of 15Z.

(2) 22 is a factor of 16Z.
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by Anindya Madhudor » Tue Dec 18, 2012 7:23 am
Statement I:
22=2*11
15z= 3*5*z
15 and 22 do not have any common factors.
So, if 22 is a factor of 15Z, 22 has to be a factor of z.
Sufficient.

Statement II:
Same logic as before.
In this case, 22 and 16 have 2 as a common factor. So, 22 may or may not be a factor of z. For example, if z=11, then 22 will be a factor of 16Z, but not of z. When z is 22, 22 is a factor of both z and 16z.
Not sufficient.
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by Brent@GMATPrepNow » Tue Dec 18, 2012 7:58 am
theachiever wrote:If Z is an integer, is 22 a factor of Z?

(1) 22 is a factor of 15Z.

(2) 22 is a factor of 16Z.
Preamble: A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Examples:
24 is divisible by 3 <--> 24 = 2x2x2x3
70 is divisible by 5 <--> 70 = 2x5x7
330 is divisible by 6 <--> 330 = 2x3x5x11
56 is divisible by 8 <--> 56 = 2x2x2x7

Target question: Is 22 a factor of Z?
Rephrased target question: Is Z divisible by 22?
Rephrased target question: Is 22 "hiding" in the prime factorization of Z?
Rephrased target question: Does the prime factorization of Z include 2 and 11?
Let's go with the last rephrased target question.

Statement 1: 22 is a factor of 15Z
In other words, 22 is "hiding" in the prime factorization of 15Z
Since 15 = (3)(5), we can say that the prime factorization of (3)(5)Z includes a 2 and an 11
From this we can conclude that the prime factorization of Z must include a 2 and an 11
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 22 is a factor of 16Z.
In other words, 22 is "hiding" in the prime factorization of 16Z
Since 16 = (2)(2)(2)(2), we can say that the prime factorization of (2)(2)(2)(2)Z includes a 2 and an 11
This leads us to several possible cases. Here are two:
Case a: Z = (2)(11), in which case the prime factorization of Z includes a 2 and an 11
Case b: Z = 11, in which case the prime factorization of Z does not include a 2 and an 11
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent
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