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If 4 is a factor of positive integer

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by NandishSS » Wed Jan 11, 2017 6:37 am
If 4 is a factor of positive integer x and 9 is a factor of positive integer y, is 42 a factor of xy?

(1) 14 is a factor of x.
(2) 25 is a factor of y.

OA:A

Source:Math Revolution

Hi Experts,

Can somebody explain how to deal with these kind of problems?

Thanks
Nandish
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Source: — Data Sufficiency |
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by melguy » Wed Jan 11, 2017 7:14 am
4 is a factor of positive integer x and 9 is a factor of positive integer y

Rephrase:
x contains a 2 x 2
y contains a 3 x 3
Question is asking does x y contains a 42 (2 x 3 x 7)?

Statement 1

x contains 2 x 7
From the question stem we know that y already contains 3. So we can say with certainty that x y contains at least one 2 x 3 x 7.

Sufficient

Statement 2

y contains 5 x 5.

It gives us no useful information about 2, 3 or 7. Insufficient.

Answer A
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by Danny@GMATAcademy » Wed Jan 11, 2017 11:33 am
We can rewrite x as 4m, and y as 9n, where m and n are integers.

The question is:

Will 42 divide evenly into xy? Or...
Will the denominator of xy/(42) cancel out completely? Or...
Will the denominator of xy/(2*3*7) cancel out completely? Or...
Will the denominator of (4m*9n)/(2*3*7) cancel out completely? Or...
Will the denominator of (2*2m*3*3n)/(2*3*7) cancel out completely? Or...
Will the denominator of (2m*3n)/(7) cancel out completely? Or...
Can we ever get that 7 out of the denominator? Or...
Does either x or y have a 7 somewhere in its Prime Factorization?

Statement 1) If 14 is a factor of x, then 7 is definitely a factor of x. Sufficient.


Statement 2) We have no idea whether either x or y has a 7 somewhere in its Prime Factorization. Insufficient.
Last edited by Danny@GMATAcademy on Wed Jan 11, 2017 1:27 pm, edited 1 time in total.
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by Brent@GMATPrepNow » Wed Jan 11, 2017 12:01 pm
NandishSS wrote:If 4 is a factor of positive integer x and 9 is a factor of positive integer y, is 42 a factor of xy?

(1) 14 is a factor of x.
(2) 25 is a factor of y.
Target question: Is 42 a factor of xy?

ASIDE----------------------------------------------
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

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
----------------------------------------------
Since 42 = (2)(3)(7), the target question can be REPHRASED as...
REPHRASED target question: Is there a 2, and 3 and a 7 hiding in the prime factorization of xy?

Given: 4 is a factor of positive integer x and 9 is a factor of positive integer y
If 4 is a factor of x, then x = 4k for some integer k
If 9 is a factor of y, then y = 9j for some integer j
So, xy = (4k)(9j) = (2)(2)(3)(3)jk (for some integer jk)
Notice that xy already (before we even examine each statement) has a 2 and a 3 hiding in its prime factorization.
The ONLY missing piece is a 7.
So, IF we get some information that tells us that there's a 7 hiding in the prime factorization of xy, then we can be certain that xy is divisible by 42.
This means we can REPHRASE our target question even more....
RE-REPHRASED target question: Is there a 7 hiding in the prime factorization of xy?

Statement 1: 14 is a factor of x
Since 14 = (2)(7), we know that there's a 2 and a 7 hiding in the prime factorization of x.
This also means there's a 2 and a 7 hiding in the prime factorization of xy.
More importantly, there's a 7 hiding in the prime factorization of xy.
Since we can answer the RE-REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: 25 is a factor of y
Since 25 = (5)(5), we know that there are two 5's hiding in the prime factorization of y.
IMPORTANT: This does not mean there are no 7's hiding in the prime factorization of y.
We just can't be sure.
As such, we cannot definitively say whether or not there are any 7's hiding in the prime factorization of xy
Since we cannot answer the RE-REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Jeff@TargetTestPrep » Mon Jan 16, 2017 5:25 pm
NandishSS wrote:If 4 is a factor of positive integer x and 9 is a factor of positive integer y, is 42 a factor of xy?

(1) 14 is a factor of x.
(2) 25 is a factor of y.
We are given that 4 is a factor of positive integer x and 9 is a factor of positive integer y. Thus:

x/4 = integer

AND

y/9 = integer

So, x is a multiple of 4 and y is a multiple of 9.

We need to determine whether xy/42 = integer. Notice that prime factorization of 42 = 2 x 3 x 7.

Statement One Alone:

14 is a factor of x.

Using the information in statement one and the stem, we know that x is a multiple of both 14 and 4, and thus x is a multiple of 28, the LCM of 14 and 4. Since x is a multiple of 28, we can let x = 28. Since we also know that y is a multiple of 9, we can let y = 9.

Finally, since (28 x 9)/42 is an integer, and since x can be any multiple of 28 and y can be any multiple of 9, we can determine that xy/42 is an integer. Statement one alone is sufficient to answer the question. We can eliminate answers B, C, and E.

Statement Two Alone:

25 is a factor of y.

Using the information in statement two and the stem, we know that y is a multiple of both 9 and 25, and thus y is a multiple of 225, the LCM of 9 and 25. We also know that x is a multiple of 4.

If we let x = 4 and y = 225, then (4 x 225)/42 is NOT an integer. Notice that 4 x 225 = 2^2 x 3^2 x 5^2 does not have a prime factor 7, but 42 = 2 x 3 x 7 does. However, if we let x = 28 and y = 225, (28 x 225)/42 IS an integer. Notice that 28 x 225 = 2^2 x 7 x 3^2 x 5^2 does have a prime factor 7. Since xy may or may not be a multiple of 42, statement two alone is not sufficient to answer the question.

Answer: A

Jeffrey Miller
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by Matt@VeritasPrep » Wed Jan 18, 2017 6:08 pm
The trick is finding a way to write these statements algebraically.

"4 is a factor of positive integer x" =>

"x is a positive multiple of 4" =>

x = 4 * some integer =>

x = 4n (where n stands for some integer)

For the second one, we've got y = 9m, where m is some positive integer.

Together, that gives us

x * y = 4n * 9m = 36 * m * n

For 36mn to be a multiple of 42, we need it to divide by 7. (36 * 7 = 252 = 42 * 6, so that divides by 42.) That gives us a nicer way of phrasing the question:

"Is either x or y divisible by 7?"

S1 says yes, x is, S2 doesn't tell us, so A.
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