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

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Veritas Prep GMAT Instructor
Joined: 01 Jul 2017
Posts: 73
Location: United States
If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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08 Jan 2018, 22:01
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Difficulty:

65% (hard)

Question Stats:

60% (02:03) correct 40% (01:55) wrong based on 93 sessions

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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

_________________
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.
Veritas Prep GMAT Instructor
Joined: 01 Jul 2017
Posts: 73
Location: United States
Re: If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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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.

_________________
Aaron J. Pond
Veritas Prep Elite-Level Instructor

Hit "+1 Kudos" if my post helped you understand the GMAT better.
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##### General Discussion
Math Expert
Joined: 02 Aug 2009
Posts: 7763
If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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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..
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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Joined: 26 Mar 2013
Posts: 2284
Re: If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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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.

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?

Veritas Prep GMAT Instructor
Joined: 01 Jul 2017
Posts: 73
Location: United States
If x and y are both positive integers, is y a multiple of 7?  [#permalink]

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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?

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?   [#permalink] 17 Mar 2018, 11:29
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