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DS(Number System)

by rintoo22 » Wed Mar 27, 2013 2:16 pm
If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
(1) x is a multiple of 9.
(2) y is a multiple of 25.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

Sol :
factors of 105 : 7,3,5
therefore Statement 1 and Statement 2 individually wont solve the issue.
When we put the 2 conditions together then the resultant number is multiple of 105.
Therefore the answer is C. However in QA then answer given is B.
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by srcc25anu » Wed Mar 27, 2013 3:05 pm
B alone is sufficient.

given X = 2*3k
Y = 2*7k

cond B states Y = multiple of 25
hence y = 5^2 * 2 * 7k
given that x = 2*3k
hence xy = 2^2 * 3 * 5^2 * 7K
105 has the following prime factors: 3,5,7

XY above is divisible by all these factors hence XY is divisible by 105
B is sufficient
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by Brian@VeritasPrep » Wed Mar 27, 2013 3:42 pm
Hey rintoo,

One thing you probably overlooked here is something the GMAT does frequently as questions get harder - they "hide" information in plain sight in the question stem. Here the question stem already says:

y is a multiple of 14 (factors of 2 and 7)
x is a multiple of 6 (factors of 2 and 3)

And you need the factors of 105 (3*5*7). You already have 3 and 7 from the question stem - but since our eyes get drawn so quickly to the statements it's easy to overlook that fact.

The lesson: Make sure you take inventory of any "hidden gem" information that you already know from the question stem!
Brian Galvin
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Chief Academic Officer
Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
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by Brent@GMATPrepNow » Wed Mar 27, 2013 3:53 pm
rintoo22 wrote:If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
(1) x is a multiple of 9.
(2) y is a multiple of 25.
Target question: Is xy a multiple of 105?

Important stuff:
First, If N is a multiple of k, then N is divisible by k.

Second, 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


Since 105 = (3)(5)(7), then we can rewrite the target question as . . .
Rephrased target question: Is there a 3, a 5 and a 7 hiding in the prime factorization of xy?

Given: x is a multiple of 6
In other words, x = (2)(3)(other possible prime numbers)

Given: y is a multiple of 14
In other words,y = (2)(7)(other possible prime numbers)

Combine both of the above to see that xy = (2)(2)(3)(7)(other possible prime numbers)

So, the given information tells us that we ALREADY have a 3 and a 7 hiding in the prime factorization of xy. The only piece missing is the 5.

So, we can rephrase our target question one last time. . .

Rephrased target question: Is there a 5 hiding in the prime factorization of xy?

Now we can check the statements.

Statement 1: x is a multiple of 9.
Since 9 = (3)(3), all this tells us is that there are two 3's hiding in the prime factorization of xy.
So, there may or may not be a 5 hiding in the prime factorization of xy.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y is a multiple of 25.
Since 25 = (5)(5), this tells us is that there is definitely a 5 hiding in the prime factorization of xy.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
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by hutch27 » Thu Mar 28, 2013 9:33 am
That is a really good explanation, Brent. I agree that integer property question do have a lot to do with prime numbers. I think the tricky part is rephrasing the target question. I'm going to tackle a few more to practice applying prime factorization techniques.



Brent@GMATPrepNow wrote:
rintoo22 wrote:If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
(1) x is a multiple of 9.
(2) y is a multiple of 25.
Target question: Is xy a multiple of 105?

Important stuff:
First, If N is a multiple of k, then N is divisible by k.

Second, 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


Since 105 = (3)(5)(7), then we can rewrite the target question as . . .
Rephrased target question: Is there a 3, a 5 and a 7 hiding in the prime factorization of xy?

Given: x is a multiple of 6
In other words, x = (2)(3)(other possible prime numbers)

Given: y is a multiple of 14
In other words,y = (2)(7)(other possible prime numbers)

Combine both of the above to see that xy = (2)(2)(3)(7)(other possible prime numbers)

So, the given information tells us that we ALREADY have a 3 and a 7 hiding in the prime factorization of xy. The only piece missing is the 5.

So, we can rephrase our target question one last time. . .

Rephrased target question: Is there a 5 hiding in the prime factorization of xy?

Now we can check the statements.

Statement 1: x is a multiple of 9.
Since 9 = (3)(3), all this tells us is that there are two 3's hiding in the prime factorization of xy.
So, there may or may not be a 5 hiding in the prime factorization of xy.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y is a multiple of 25.
Since 25 = (5)(5), this tells us is that there is definitely a 5 hiding in the prime factorization of xy.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
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