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If x and y are both positive integers, is y a multiple of 7?

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If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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New post 08 Jan 2018, 22:01
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If \(x\) and \(y\) are both positive integers, is \(y\) a multiple of 7?

(1) \((x^3)(y^3) = 441^3\)
(2) \(x\) is a single digit integer

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Re: If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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New post 08 Jan 2018, 22:39
Statement #1 tells us that \((x^3)(y^3) = 441^3\). This can be mathematically simplified down to \((x)(y)=441\). Therefore, both \(x\) and \(y\) are factors of \(441\).

Factoring \(441\) isn't hard, especially once we realize that \(441\) is not only divisible by \(9\), but \(441 = 450-9\).

Factoring a common \(9\) out of this expression gives us:

\(450-9 = 9(50-1) = 9*49 = 3*3*7*7\)

\(x\) or \(y\) could be any combination of these prime values. \(y\) could be equal to 3 (not a multiple of 7), and it could also be equal to 7 (obviously, a multiple of 7.) Statement #1 is clearly insufficient.

Statement #2 is profoundly insufficient by itself. It tells us nothing about \(y\), so it is easy to eliminate.

As we combine the statements together, we can now ask ourselves the question, "does it matter that \(x\) is a single digit integer?" If \(x\) must be a single digit integer, there are only four possible solutions: \(x\)\(=1,3,7,\) or \(9\).

Given these four values for \(x\), here are the possible values for \(x\) and \(y\) (notice that there is no reason to actually multiply them out, since the problem only wants us to find out if \(y\) is a multiple of 7.)

(\(x\)) *(\(y\))
\(1\) * (\(3*3*7*7\))
\(3\)*(\(3*7*7\))
\(7\)*(\(3*3*7\))
\(9\)*(\(7*7\))

In every case, \(y\) must be a multiple of 7. Combining these two statements together is sufficient.

The answer is C.
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If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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New post 08 Jan 2018, 22:26
AaronPond wrote:
If \(x\) and \(y\) are both positive integers, is \(y\) a multiple of 7?

(1) \((x^3)(y^3) = 441^3\)
(2) \(x\) is a single digit integer



Points to consider..
1) x and y are positive integers..

From statement I,
\(x*y = 441=1*3*3*7*7\)
So x can be 9, and y can be 49.... Yes y is multiple of 7
x can be 49 and y can be 9..... Ans NO
Insufficient

From statement II..
Nothing about y..
Insufficient

Combined...
x can be 1,3,7 or 9
corresponding values of y are 441,3*49,63,49
In each case y will be MULTIPLE of 7
Sufficient
C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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New post 16 Mar 2018, 16:10
AaronPond wrote:
Statement #1 tells us that \((x^3)(y^3) = 441^3\). This can be mathematically simplified down to \((x)(y)=441\). Therefore, both \(x\) and \(y\) are factors of \(441\).

Factoring \(441\) isn't hard, especially once we realize that \(441\) is not only divisible by \(9\), but \(441 = 450-9\).

Factoring a common \(9\) out of this expression gives us:

\(450-9 = 9(50-1) = 9*49 = 3*3*7*7\)

\(x\) or \(y\) could be any combination of these prime values. \(y\) could be equal to 3 (not a multiple of 7), and it could also be equal to 7 (obviously, a multiple of 7.) Statement #1 is clearly insufficient.

Statement #2 is profoundly insufficient by itself. It tells us nothing about \(y\), so it is easy to eliminate.

As we combine the statements together, we can now ask ourselves the question, "does it matter that \(x\) is a single digit integer?" If \(x\) must be a single digit integer, there are only four possible solutions: \(x\)\(=1,3,7,\) or \(9\).

Given these four values for \(x\), here are the possible values for \(x\) and \(y\) (notice that there is no reason to actually multiply them out, since the problem only wants us to find out if \(y\) is a multiple of 7.)

(\(x\)) *(\(y\))
\(1\) * (\(3*3*7*7\))
\(3\)*(\(3*7*7\))
\(7\)*(\(3*3*7\))
\(9\)*(\(7*7\))

In every case, \(y\) must be a multiple of 7. Combining these two statements together is sufficient.

The answer is C.


Dear AaronPond

In evaluating statement 1 alone, Can I say the following:

(x) (y) = 441

(441) (1) = 441

(1) (441) = 441

Does it have to be inform of prime for statement 1?

Thanks in advance
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Joined: 01 Jul 2017
Posts: 49
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If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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New post 17 Mar 2018, 11:29
Mo2men wrote:

Dear AaronPond

In evaluating statement 1 alone, Can I say the following:

(x) (y) = 441

(441) (1) = 441

(1) (441) = 441

Does it have to be inform of prime for statement 1?

Thanks in advance


Great question! No, the problem does not require your factorization to be in the form of a prime. That is why I said "\(x\) or \(y\) could be any combination of these prime values." They both do not need to be a combination of primes. (Remember: "\(1\)" is not a prime number. Since the definition of a prime is "a number divisible by only two distinct factors: 1 and itself", \(2\) is the first prime number.)

You can also see this idea embedded in my explanation above. One of the possible options for \(x*y\) is \(1\) * (\(3*3*7*7\))

Keep studying smart and asking solid questions!
_________________

Aaron J. Pond
Veritas Prep Elite-Level Instructor

Hit "+1 Kudos" if my post helped you understand the GMAT better.
Look me up at https://www.veritasprep.com/gmat/aaron-pond/ if you want to learn more GMAT Jujitsu.

If x and y are both positive integers, is y a multiple of 7? &nbs [#permalink] 17 Mar 2018, 11:29
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