Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 21 Dec 2013, 04:18

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x and y are positive integers and x/y has a remainder of

 Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
Manager
Joined: 04 Dec 2008
Posts: 114
Followers: 1

Kudos [?]: 13 [0], given: 2

If x and y are positive integers and x/y has a remainder of [#permalink]  20 Jan 2010, 19:54
00:00

Difficulty:

5% (low)

Question Stats:

75% (02:25) correct 25% (00:00) wrong based on 4 sessions
If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

[Reveal] Spoiler:
This ques is from Man prep book. I don't understand why smallest value of x is 5 and not 11.
 Kaplan Promo Code Knewton GMAT Discount Codes Manhattan GMAT Discount Codes
Math Expert
Joined: 02 Sep 2009
Posts: 15220
Followers: 2563

Kudos [?]: 15863 [3] , given: 1575

Re: Remainder of 5 [#permalink]  01 Feb 2010, 04:17
3
KUDOS
Expert's post
x=qy+5, not y+5, and note that q can be zero. In this case x=0*y+5=5, this is smallest value of x.

Hey Bunuel....you lose me after the last part above... I still don't see how y=6. Could you explain step by step? or does anyone have a solution that doesn't involve algebra?

\frac{5}{y} gives remainder 5 means y>x. For EVERY y more than x=5, \frac{5}{y} will give remainder of 5: \frac{5}{6}, \frac{5}{7}, \frac{5}{16778}, ... all these fractions have remainder of 5.

I thought \frac{5}{6}, is equal to .8 with a remainder of 2?

THEORY:
If a and d are positive integers, there exists unique integers q and r, such that a = qd + r and 0\leq{r}<d. Where a is a dividend, d is a divisor, q is a quotient (\geq0) and r is a remainder.

Now consider the case when dividend a is less than divisor d. For instance: what is the ramainder when a=7 divided by d=9:

a = qd + r --> 7=q*9+r, as q\geq0 and 0\leq{r}<d --> 7=0*9+r=0*9+7 --> r=remainder=7.

Hence in ANY case when positive integer a is divided by positive integer d and given that a<d, remainder will always be a. Note here that EVERY GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

So: \frac{7}{9}, remainder 7; \frac{11}{13}, remainder 11; \frac{135}{136}, remainder 135.

Back to our original question:

When we got that x=5 and \frac{5}{y} gives remainder of 5, according to the above we can conclude that y must be more than x=5. As we want to minimize the x*y we need least integer y which is more than x=5. Thus y=6.

Check it:
5=0*6+5 --> x=5 dividend, y=6 divisor, q=0 quotient and r=5 remainder.

Hope it helps.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 15220
Followers: 2563

Kudos [?]: 15863 [1] , given: 1575

Re: Remainder of 5 [#permalink]  21 Jan 2010, 00:49
1
KUDOS
Expert's post
joyseychow wrote:
If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

[spoiler]This ques is from Man prep book. I don't understand why smallest value of x is 5 and not 11.[/spoiler]

Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y.

Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 15220
Followers: 2563

Kudos [?]: 15863 [1] , given: 1575

Re: Remainder of 5 [#permalink]  30 Jan 2010, 14:59
1
KUDOS
Expert's post
jeeteshsingh wrote:
Bunuel wrote:
Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y.

Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30.

How do you manage to do such things???? What is the best approach to master such things??? Please guide!

What "things" exactly do you mean?

This question is about the remainders, so would suggest to review this concept in guids and/or to practice as much as possible these types of qestions (refer to the tags list by forum: REMAINDERS). Also you can check sriharimurthy's topic about the remainders at: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Hope it helps.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 15220
Followers: 2563

Kudos [?]: 15863 [1] , given: 1575

Re: Remainder of 5 [#permalink]  31 Jan 2010, 07:43
1
KUDOS
Expert's post
jeeteshsingh wrote:
Bunuel wrote:

Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y.

Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30.

Bunuel... am again lost here...

As said x = qy + 5...... and since q\geq0... this should be mean that X = Y + 5... then how come X can be less than Y.... we are adding 5 to Y to get X...

x=qy+5, not y+5, and note that q can be zero. In this case x=0*y+5=5, this is smallest value of x.

\frac{5}{y} gives remainder 5 means y>x. For EVERY y more than x=5, \frac{5}{y} will give remainder of 5: \frac{5}{6}, \frac{5}{7}, \frac{5}{16778}, ... all these fractions have remainder of 5.

Hope it's clear.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 3781
Location: Pune, India
Followers: 813

Kudos [?]: 3244 [1] , given: 138

Re: Remainder question [#permalink]  02 Aug 2012, 22:32
1
KUDOS
Expert's post
GmatSlayer112 wrote:
Can someone help me understand this?

The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy?

The answer is 5*6 = 30

I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible.

Think: What is the quotient and what is the remainder when 3 is divided by 10?
The quotient is 0 (an integer) and the remainder is 3.

What is the quotient and what is the remainder when 5 is divided by 12?
The quotient is 0 (an integer) and the remainder is 5.

When a smaller number is divided by a larger number, the quotient is 0 and the remainder is the smaller number.

Divisibility is basically grouping. When you divide n by 10, you make as many groups of 10 as you can and the leftover is the remainder. When you have 3 and you want to divide it by 10, you make 0 groups of 10 each and you have 3 leftover which is the remainder.
See this post: http://www.veritasprep.com/blog/2011/04 ... unraveled/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Intern Joined: 20 Jan 2010 Posts: 1 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Remainder of 5 [#permalink] 20 Jan 2010, 21:32 As the remainder is 5 the numerator has to be a multiple of 5. Since only multiples of 5 have exactly 5 units between each one of them(5,10,15,20..). Now to tackle the denominator...look for the smallest possible positive integer. If we take 1, we quickly realize that it completely divides any number (remainder is 0). So we go to the next integer which is 2 and we can conclude the multiple of 5 that should be in the numerator is 5x2=10. Thus we can see that the smallest possible value is 10 x 2=20. Senior Manager Joined: 22 Dec 2009 Posts: 366 Followers: 9 Kudos [?]: 167 [0], given: 47 Re: Remainder of 5 [#permalink] 30 Jan 2010, 14:45 Bunuel wrote: Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y. Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30. How do you manage to do such things???? What is the best approach to master such things??? Please guide! _________________ Cheers! JT........... If u like my post..... payback in Kudos!! |Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice| |For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide| ~~Better Burn Out... Than Fade Away~~ Senior Manager Joined: 22 Dec 2009 Posts: 366 Followers: 9 Kudos [?]: 167 [0], given: 47 Re: Remainder of 5 [#permalink] 31 Jan 2010, 07:26 Bunuel wrote: Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y. Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30. Bunuel... am again lost here... As said x = qy + 5...... and since q\geq0... this should be mean that X = Y + 5... then how come X can be less than Y.... we are adding 5 to Y to get X... _________________ Cheers! JT........... If u like my post..... payback in Kudos!! |Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice| |For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide| ~~Better Burn Out... Than Fade Away~~ Intern Joined: 12 Oct 2009 Posts: 16 Followers: 1 Kudos [?]: 2 [0], given: 1 Re: Remainder of 5 [#permalink] 31 Jan 2010, 19:02 Bunuel wrote: jeeteshsingh wrote: Bunuel wrote: Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y. Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30. Bunuel... am again lost here... As said x = qy + 5...... and since q\geq0... this should be mean that X = Y + 5... then how come X can be less than Y.... we are adding 5 to Y to get X... x=qy+5, not y+5, and note that q can be zero. In this case x=0*y+5=5, this is smallest value of x. Hey Bunuel....you lose me after the last part above... I still don't see how y=6. Could you explain step by step? or does anyone have a solution that doesn't involve algebra? \frac{5}{y} gives remainder 5 means y>x. For EVERY y more than x=5, \frac{5}{y} will give remainder of 5: \frac{5}{6}, \frac{5}{7}, \frac{5}{16778}, ... all these fractions have remainder of 5. I thought \frac{5}{6}, is equal to .8 with a remainder of 2? Manager Joined: 10 Feb 2010 Posts: 197 Followers: 2 Kudos [?]: 25 [0], given: 6 Re: Remainder of 5 [#permalink] 12 Feb 2010, 17:36 If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy? y>5 and smallest number greater than 5 is 6. x=y(quotient)+reminder = 6(quotient)+5 to get smallest number, substitute 0 =>x=6(0)+5=5 and xy=5x6=30 Intern Joined: 06 Aug 2010 Posts: 9 Followers: 0 Kudos [?]: 0 [0], given: 0 MGMAT Remainder Question [#permalink] 18 Aug 2010, 16:31 i need a better explanation for their solution please. Problem: If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy? solution: 30: The remainder must always be smaller than the divisor. In this problem, 5 must be smaller than y. Additionally, y must be an integer, so y must be at least 6. If y is 6, then the smallest possible value of x is 5. (Other values of x that leave a remainder of 5 when divided by 6 would be 11, 17, 23, etc.) If y is chosen to be larger than 6, then the smallest possible value of x is still 5. Thus, we will get the smallest possible value of the product xy by choosing the smallest x together with the smallest y. The smallest possible value of xy is 5X6=30. My problem, is that I do not know why the smallest possible value of x is 5. My logic is that since the remainder is 5, y is greater than 5 and is in turn 6. So for a number to yield a remainder of 5 after being divided by 6, 5+6=11. X must be 11 and xy=66. Dont see how you can have 5/6 has a reminader of 5. thanks Senior Manager Status: Time to step up the tempo Joined: 24 Jun 2010 Posts: 416 Location: Milky way Schools: ISB, Tepper - CMU, Chicago Booth, LSB Followers: 6 Kudos [?]: 85 [0], given: 50 Re: MGMAT Remainder Question [#permalink] 18 Aug 2010, 18:04 chsebik wrote: i need a better explanation for their solution please. Problem: If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy? solution: 30: The remainder must always be smaller than the divisor. In this problem, 5 must be smaller than y. Additionally, y must be an integer, so y must be at least 6. If y is 6, then the smallest possible value of x is 5. (Other values of x that leave a remainder of 5 when divided by 6 would be 11, 17, 23, etc.) If y is chosen to be larger than 6, then the smallest possible value of x is still 5. Thus, we will get the smallest possible value of the product xy by choosing the smallest x together with the smallest y. The smallest possible value of xy is 5X6=30. My problem, is that I do not know why the smallest possible value of x is 5. My logic is that since the remainder is 5, y is greater than 5 and is in turn 6. So for a number to yield a remainder of 5 after being divided by 6, 5+6=11. X must be 11 and xy=66. Dont see how you can have 5/6 has a reminader of 5. thanks Quote: So for a number to yield a remainder of 5 after being divided by 6 Think about this. If x is 5 and y is 6, then when 5 is divided by 6, the quotient is 0 and the remainder is 5. _________________ Support GMAT Club by putting a GMAT Club badge on your blog Intern Joined: 13 Nov 2011 Posts: 30 Location: United States GMAT Date: 05-26-2012 GPA: 3.2 WE: Supply Chain Management (Health Care) Followers: 0 Kudos [?]: 5 [0], given: 1 Remainder question [#permalink] 02 Aug 2012, 16:45 Can someone help me understand this? The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy? The answer is 5*6 = 30 I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible. Kaplan GMAT Instructor Joined: 25 Aug 2009 Posts: 644 Location: Cambridge, MA Followers: 62 Kudos [?]: 157 [0], given: 2 Re: Remainder question [#permalink] 02 Aug 2012, 18:17 Expert's post GmatSlayer112 wrote: Can someone help me understand this? The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy? The answer is 5*6 = 30 I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible. if y is 5, then 5/6 = 0 r 5 -- that is, there are NO complete sixes in the number five, so all five are left over in the remainder! _________________ Eli Meyer Kaplan Teacher http://www.kaptest.com/GMAT Prepare with Kaplan and save$150 on a course!

Kaplan Reviews

Director
Joined: 22 Mar 2011
Posts: 609
WE: Science (Education)
Followers: 59

Kudos [?]: 371 [0], given: 43

Re: Remainder question [#permalink]  02 Aug 2012, 22:08
GmatSlayer112 wrote:
Can someone help me understand this?

The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy?

The answer is 5*6 = 30

I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible.

When dividing a positive integer x by another positive integer y, we get a unique non-negative integer quotient q and a unique remainder r, where r is an integer such that 0\leq{r}<y. When r = 0, we say that x is divisible by y, that y is a factor of x, or that x is a multiple of y. In this case, x can be evenly divided by y.

In our case, we can write x = q*y + 5, and as the remainder is always smaller than the divisor, we can deduce that y should be at least 6. The minimum quotient q is 0 which gives x = 5. Therefore, the smallest possible value of xy is 5 * 6 =30.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Math Expert
Joined: 02 Sep 2009
Posts: 15220
Followers: 2563

Kudos [?]: 15863 [0], given: 1575

Re: Remainder question [#permalink]  02 Aug 2012, 23:26
Expert's post
GmatSlayer112 wrote:
Can someone help me understand this?

The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy?

The answer is 5*6 = 30

I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible.

Merging similar topics. Please refer to the solutions above.
_________________
Intern
Joined: 13 Nov 2011
Posts: 30
Location: United States
GMAT Date: 05-26-2012
GPA: 3.2
WE: Supply Chain Management (Health Care)
Followers: 0

Kudos [?]: 5 [0], given: 1

Re: Remainder question [#permalink]  04 Aug 2012, 09:32
VeritasPrepKarishma wrote:
GmatSlayer112 wrote:
Can someone help me understand this?

The question is if x and y are positive integers and x/y has a remainder of 5, what is the smallest possible of xy?

The answer is 5*6 = 30

I came up with 66 because I understand that y has to be at least 6 and with that I would think that y has to be 11 which is how I derived 66. The answer says that the smallest y is 5 but I don't understand how that is possible.

Think: What is the quotient and what is the remainder when 3 is divided by 10?
The quotient is 0 (an integer) and the remainder is 3.

What is the quotient and what is the remainder when 5 is divided by 12?
The quotient is 0 (an integer) and the remainder is 5.

When a smaller number is divided by a larger number, the quotient is 0 and the remainder is the smaller number.

Divisibility is basically grouping. When you divide n by 10, you make as many groups of 10 as you can and the leftover is the remainder. When you have 3 and you want to divide it by 10, you make 0 groups of 10 each and you have 3 leftover which is the remainder.
See this post: http://www.veritasprep.com/blog/2011/04 ... unraveled/

Thanks very helpful, especially the blog post.
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 238
Schools: Johnson '15
Followers: 2

Kudos [?]: 19 [0], given: 16

Re: Remainder of 5 [#permalink]  04 Aug 2012, 17:33
Bunuel wrote:
joyseychow wrote:
If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

[spoiler]This ques is from Man prep book. I don't understand why smallest value of x is 5 and not 11.[/spoiler]

Given x=qy+5, where q is a quotient, an integer \geq0. Which means that the least value of x is when q=0, in that case x=5. This basically means that numerator x, is less than denominator y.

Now the smallest denominator y, which is more than numerator x=5 is 6. so we have x=5 and y=6 --> xy=30.

for this first time, i exactly though the same way as Bunuel...good teacher bunuel...thanks for this,,,,
_________________

Regards,
Harsha

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Satyameva Jayate - Truth alone triumphs

Re: Remainder of 5   [#permalink] 04 Aug 2012, 17:33
Similar topics Replies Last post
Similar
Topics:
If x and y are positive integers, what is the remainder when 7 10 Jan 2005, 11:26
4 If x and y are positive integers, what is the remainder when 9 22 Apr 2005, 09:46
X/Y has a unknown remainder. Both X and Y are positive 2 25 Oct 2007, 05:54
If X and Y are positive integers, what is the remainder of 1 07 Nov 2007, 08:53
If X and Y are positive integers, what is the remainder of 9 18 Dec 2007, 10:25
Display posts from previous: Sort by

# If x and y are positive integers and x/y has a remainder of

 Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.