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# If x and y are positive integers, what is the value of x?

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Joined: 11 Sep 2015
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GMAT 1: 770 Q49 V46
If x and y are positive integers, what is the value of x?  [#permalink]

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Updated on: 27 Mar 2017, 09:14
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If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

*kudos for all correct solutions

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Originally posted by BrentGMATPrepNow on 27 Mar 2017, 06:44.
Last edited by BrentGMATPrepNow on 27 Mar 2017, 09:14, edited 1 time in total.
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 07:12
1
2
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

*kudos for all correct solutions

Hi

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5

9x = 2y*x + 5

9x - 2yx = 5

x(9 - 2y) = 5

5 is prime so we can have only two choices 1*5 or 5*1.

x = 1, 9 - 2y = 5 ---> y = 2 Checking: 9*1/2*2 = 9/4 remainder 1 Discard.

x = 5, 9 - 2y = 1 ---> y = 4. Checking: 9*5/2*4 = 45/8 remainder 5. Sufficient.

2) When 5y is divided by x, the quotient is y, and the remainder is 0

5y = xy

y(5 - x) = 0 -----> either y = 0 or x = 5. But when y=0 x can take any value. Insufficient.

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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 08:08
2
vitaliyGMAT wrote:
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

*kudos for all correct solutions

Hi

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5

9x = 2y*x + 5

9x - 2yx = 5

x(9 - 2y) = 5

5 is prime so we can have only two choices 1*5 or 5*1.

x = 1, 9 - 2y = 5 ---> y = 2 Checking: 9*1/2*2 = 9/4 remainder 1 Discard.

x = 5, 9 - 2y = 1 ---> y = 4. Checking: 9*5/2*4 = 45/8 remainder 5. Sufficient.

2) When 5y is divided by x, the quotient is y, and the remainder is 0

5y = xy

y(5 - x) = 0 -----> either y = 0 or x = 5. But when y=0 x can take any value. Insufficient.

y cannot be 0 as the question says y is a positive integer.

I think the answer should be D.

Cheers!
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Joined: 11 Sep 2015
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If x and y are positive integers, what is the value of x?  [#permalink]

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Updated on: 27 Mar 2017, 09:19
Top Contributor
1
Diwakar003 wrote:
vitaliyGMAT wrote:
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

*kudos for all correct solutions

Hi

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5

9x = 2y*x + 5

9x - 2yx = 5

x(9 - 2y) = 5

5 is prime so we can have only two choices 1*5 or 5*1.

x = 1, 9 - 2y = 5 ---> y = 2 Checking: 9*1/2*2 = 9/4 remainder 1 Discard.

x = 5, 9 - 2y = 1 ---> y = 4. Checking: 9*5/2*4 = 45/8 remainder 5. Sufficient.

2) When 5y is divided by x, the quotient is y, and the remainder is 0

5y = xy

y(5 - x) = 0 -----> either y = 0 or x = 5. But when y=0 x can take any value. Insufficient.

y cannot be 0 as the question says y is a positive integer.

I think the answer should be D.

Cheers!

You are absolutely correct. I was incorrectly reading my target question as "What is the value of y?"
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Originally posted by BrentGMATPrepNow on 27 Mar 2017, 08:19.
Last edited by BrentGMATPrepNow on 27 Mar 2017, 09:19, edited 1 time in total.
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 08:45
2
Hi Brent,
Great Q...
But statement 2 is sufficient as we are looking for value of x and that is 5..
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 09:06
1
Why is the OA not B.

Here is what i did for this question =>

We need the value of x.

Statement 1 => 9x=2xy+5
This tells us that x must be odd.
Cool.Lets use some test cases.

x=1,y=2 => Works
x=5,y=4 => Works

Hence not sufficient.
Statement 2 => xy=5y
Hence either y=0 or x=5
We are told that x,y are positive integers.
Hence x must be 5

Sufficient.

Smash that B.

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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 09:17
Top Contributor
chetan2u wrote:
Hi Brent,
Great Q...
But statement 2 is sufficient as we are looking for value of x and that is 5..

Arggh!

I wrote that question a couple of weeks ago, and had the correct answer is D.
This morning, before posting it, I answered the question TWICE and got A both times.
Of course, I was incorrectly reading the target question as "What is the value of y"

I've changed the correct answer to D

Cheers and thanks,
Brent
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 09:21
1
Diwakar003 wrote:
vitaliyGMAT wrote:
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

*kudos for all correct solutions

Hi

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5

9x = 2y*x + 5

9x - 2yx = 5

x(9 - 2y) = 5

5 is prime so we can have only two choices 1*5 or 5*1.

x = 1, 9 - 2y = 5 ---> y = 2 Checking: 9*1/2*2 = 9/4 remainder 1 Discard.

x = 5, 9 - 2y = 1 ---> y = 4. Checking: 9*5/2*4 = 45/8 remainder 5. Sufficient.

2) When 5y is divided by x, the quotient is y, and the remainder is 0

5y = xy

y(5 - x) = 0 -----> either y = 0 or x = 5. But when y=0 x can take any value. Insufficient.

y cannot be 0 as the question says y is a positive integer.

I think the answer should be D.

Cheers!

Hi

In a hurry I missed the fact that y and x should be positive.

If y>0 then x - 5 should be = 0 and x = 5.

And 5y/5 will give us remainder 0 with any y.

Statement 2 is sufficient.

Unswer needs to be D.

Regards +1
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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27 Mar 2017, 12:08
nice question. is there any other simple way to solve this problem?
Thank you...
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If x and y are positive integers, what is the value of x?  [#permalink]

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Updated on: 06 Oct 2019, 06:43
Top Contributor
1
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

Target question: What is the value of x?

Statement 1: When 9x is divided by 2y, the quotient is x, and the remainder is 5
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, from the above rule, we can write: 9x = 2yx + 5
Rewrite this as: 9x - 2yx = 5
Factor out the x to get: x(9 - 2y) = 5
Since x and 9 - 2y MUST BE INTEGERS, and since their PRODUCT is 5, we know that EITHER x = 1 and 9 - 2y = 5 OR x = 5 and 9 - 2y = 1

If x = 1 and 9 - 2y = 5, then x = 1 and y = 2
If x = 5 and 9 - 2y = 1, then x = 5 and y = 4

At this point it LOOKS LIKE there are two possible values for x. However, when we test these values, we have a problem.

If we plug x = 1 and y = 2 into statement 1, we get: When 9(1) is divided by 2(2), the quotient is 2, and the remainder is 1 (but we need a remainder of 5). So, x = 1 and y = 2 is NOT A SOLUTION

If we plug x = 5 and y = 4 into statement 1, we get: When 9(5) is divided by 2(4), the quotient is 5, and the remainder is 5. In other words, When 45 is divided by 8, the quotient is 5, and the remainder is 5. This WORKS. So, x = 5 and y = 4 IS A SOLUTION

So, we can conclude that x = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When 5y is divided by x, the quotient is y, and the remainder is 0
So, from the above rule, we can write: 5y = xy + 0
Rewrite this as: 5y - xy = 0
Factor: y(5 - x) = 0
So, either y = 0, or x = 5
Since we're told y is POSITIVE, we know that y does not equal 0, which means x must equal 5
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

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Originally posted by BrentGMATPrepNow on 28 Mar 2017, 05:32.
Last edited by BrentGMATPrepNow on 06 Oct 2019, 06:43, edited 1 time in total.
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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28 Mar 2017, 09:58
Yes, Thanks for video.
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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06 Oct 2019, 06:13
GMATPrepNow wrote:
GMATPrepNow wrote:
If x and y are positive integers, what is the value of x?

1) When 9x is divided by 2y, the quotient is x, and the remainder is 5
2) When 5y is divided by x, the quotient is y, and the remainder is 0

Target question: What is the value of x?

Statement 1: When 9x is divided by 2y, the quotient is x, and the remainder is 5
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

So, from the above rule, we can write: 9x = 2yx + 5
Rewrite this as: 9x - 2yx = 5
Factor out the x to get: x(9 - 2y) = 5
Since x and 9 - 2y MUST BE INTEGERS, and since their PRODUCT is 5, we know that EITHER x = 1 and 9 - 2y = 5 OR x = 5 and 9 - 2y = 1

If x = 1 and 9 - 2y = 5, then x = 1 and y = 2
If x = 5 and 9 - 2y = 1, then x = 5 and y = 4

At this point it LOOKS LIKE there are two possible values for x. However, when we test these values, we have a problem.

If we plug x = 1 and y = 2 into statement 1, we get: When 9(1) is divided by 2(2), the quotient is 1, and the remainder is 5. We can see this this is NOT TRUE. So, x = 1 and y = 2 is NOT A SOLUTION

If we plug x = 5 and y = 4 into statement 1, we get: When 9(5) is divided by 2(4), the quotient is 5, and the remainder is 5. In other words, When 45 is divided by 8, the quotient is 5, and the remainder is 5. This WORKS. So, x = 5 and y = 4 IS A SOLUTION

So, we can conclude that x = 5
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When 5y is divided by x, the quotient is y, and the remainder is 0
So, from the above rule, we can write: 5y = xy + 0
Rewrite this as: 5y - xy = 0
Factor: y(5 - x) = 0
So, either y = 0, or x = 5
Since we're told y is POSITIVE, we know that y does not equal 0, which means x must equal 5
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

RELATED VIDEO

Hi Brent,

Is my understanding correct that x=1 and y=2 is being eliminated since the remainder 5 is greater than the divisor 2y? Or it violates the rule $$0<=r<d$$

Warm Regards,
Pritishd
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Re: If x and y are positive integers, what is the value of x?  [#permalink]

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06 Oct 2019, 06:46
Top Contributor
Pritishd wrote:
Hi Brent,

Is my understanding correct that x=1 and y=2 is being eliminated since the remainder 5 is greater than the divisor 2y? Or it violates the rule $$0<=r<d$$

Warm Regards,
Pritishd

Hi Pritishd,

You're correct.
The remainder (5) must always be less than the divisor (4).

I realize that I explained this part of my solution poorly. I've edited that part as follows:
If we plug x = 1 and y = 2 into statement 1, we get: When 9(1) is divided by 2(2), the quotient is 2, and the remainder is 1 (but we need a remainder of 5). So, x = 1 and y = 2 is NOT A SOLUTION
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Re: If x and y are positive integers, what is the value of x?   [#permalink] 06 Oct 2019, 06:46