GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

It is currently 25 May 2020, 07:48

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

When positive integer x is divided by 5, the remainder is 3

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
Joined: 15 Jun 2011
Posts: 1
When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post Updated on: 06 Jun 2013, 05:31
4
28
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

77% (02:12) correct 23% (02:37) wrong based on 528 sessions

HideShow timer Statistics

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?

A. 12
B. 15
C. 20
D. 28
E. 35

Originally posted by shopaholic on 02 Mar 2012, 09:39.
Last edited by Bunuel on 06 Jun 2013, 05:31, edited 2 times in total.
Edited the question, added the answer choices and OA
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 64102
Re: How to solve this problem  [#permalink]

Show Tags

New post 02 Mar 2012, 10:16
14
14
shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7,
the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when
y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of
x - y?

thanks in advance


Welcome to GMAT Club. Below is a solution to your question.

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?
A. 12
B. 15
C. 20
D. 28
E. 35

When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: \(x=5q+3\) (x could be 3, 8, 13, 18, 23, ...) and \(x=7p+4\) (x could be 4, 11, 18, 25, ...).

There is a way to derive general formula based on above two statements:

Divisor will be the least common multiple of above two divisors 5 and 7, hence \(35\).

Remainder will be the first common integer in above two patterns, hence \(18\) --> so, to satisfy both this conditions x must be of a type \(x=35m+18\) (18, 53, 88, ...);

The same for y (as the same info is given about y): \(y=35n+18\);

\(x-y=(35m+18)-(35n+18)=35(m-n)\) --> thus x-y must be a multiple of 35.

Answer: E.

More about this concept:
manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.

P.S. Please post answer choices for PS questions.
_________________
Most Helpful Community Reply
Current Student
avatar
B
Joined: 09 Apr 2013
Posts: 39
Location: United States (DC)
Concentration: Strategy, Social Entrepreneurship
Schools: Ross '20 (A$)
GMAT 1: 750 Q50 V41
GPA: 3.55
WE: General Management (Non-Profit and Government)
Reviews Badge
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 06 Jun 2013, 08:50
5
1
I do the "for dummies" way on this, because it is the first thing that popped in my mind, and the first thing I would have tried. It actually doesn't take that long on paper. Easily under 2 minutes.

5x+3:
5+3=8
10+3=13
15+3=18
20+3=23
25+3=28
30+3=33
35+3=38
40+3=43
45+3=48
50+3=53

7x+4:
7+4=11
14+4=18
21+4=25
28+4=32
35+4=39
42+4=46
49+4=53
56+4=60
63+4=67
70+4=74

The thing about PS problems is that there can only be one answer. So as long as you can find it one time, it will be the same for all other times.
The question is asking for the larger number minus the smaller number
so, 53-18 = 35

Answer is E
General Discussion
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 64102
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 06 Jun 2013, 05:32
1
1
Manager
Manager
avatar
Joined: 26 Sep 2013
Posts: 181
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41
GMAT 2: 730 Q49 V41
Re: How to solve this problem  [#permalink]

Show Tags

New post 23 Oct 2013, 18:45
Bunuel wrote:
shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7,
the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when
y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of
x - y?

thanks in advance


Welcome to GMAT Club. Below is a solution to your question.

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?
A. 12
B. 15
C. 20
D. 28
E. 35

When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: \(x=5q+3\) (x could be 3, 8, 13, 18, 23, ...) and \(x=7p+4\) (x could be 4, 11, 18, 25, ...).

There is a way to derive general formula based on above two statements:

Divisor will be the least common multiple of above two divisors 5 and 7, hence \(35\).

Remainder will be the first common integer in above two patterns, hence \(18\) --> so, to satisfy both this conditions x must be of a type \(x=35m+18\) (18, 53, 88, ...);

The same for y (as the same info is given about y): \(y=35n+18\);

\(x-y=(35m+18)-(35n+18)=35(m-n)\) --> thus x-y must be a multiple of 35.

Answer: E.

More about this concept:
manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.

P.S. Please post answer choices for PS questions.



there must be some other way to solve this; I have no idea how anyone that reads that question could sit there and think of what you wrote out, in under two minutes. It's a great solution, but I think there must be some other way to crack this nut.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 64102
Re: How to solve this problem  [#permalink]

Show Tags

New post 23 Oct 2013, 23:29
AccipiterQ wrote:
Bunuel wrote:
shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7,
the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when
y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of
x - y?

thanks in advance


Welcome to GMAT Club. Below is a solution to your question.

When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?
A. 12
B. 15
C. 20
D. 28
E. 35

When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: \(x=5q+3\) (x could be 3, 8, 13, 18, 23, ...) and \(x=7p+4\) (x could be 4, 11, 18, 25, ...).

There is a way to derive general formula based on above two statements:

Divisor will be the least common multiple of above two divisors 5 and 7, hence \(35\).

Remainder will be the first common integer in above two patterns, hence \(18\) --> so, to satisfy both this conditions x must be of a type \(x=35m+18\) (18, 53, 88, ...);

The same for y (as the same info is given about y): \(y=35n+18\);

\(x-y=(35m+18)-(35n+18)=35(m-n)\) --> thus x-y must be a multiple of 35.

Answer: E.

More about this concept:
manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.

P.S. Please post answer choices for PS questions.



there must be some other way to solve this; I have no idea how anyone that reads that question could sit there and think of what you wrote out, in under two minutes. It's a great solution, but I think there must be some other way to crack this nut.


If you know the trick to derive general formula you can solve the question just under 2 minutes.

Else, you can find common numbers (18 and 53) and see that 53-18 is a multiple of only 35 from answer choices.
_________________
Manager
Manager
avatar
Joined: 26 Sep 2013
Posts: 181
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41
GMAT 2: 730 Q49 V41
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 24 Oct 2013, 08:01
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199


Actually, on this problem, if 5 is a factor of it, as is 7, couldn't you just look at the stem, and multiply 5 by 7 without doing any extra work, to get the answer?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 64102
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 24 Oct 2013, 08:04
AccipiterQ wrote:
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on remainders problems: remainders-144665.html

All DS remainders problems to practice: search.php?search_id=tag&tag_id=198
All PS remainders problems to practice: search.php?search_id=tag&tag_id=199


Actually, on this problem, if 5 is a factor of it, as is 7, couldn't you just look at the stem, and multiply 5 by 7 without doing any extra work, to get the answer?


No. Neither 5 nor 7 is a factor of either x or y.
_________________
Veritas Prep GMAT Instructor
User avatar
V
Joined: 16 Oct 2010
Posts: 10442
Location: Pune, India
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 25 Nov 2014, 20:30
2
AccipiterQ wrote:
there must be some other way to solve this; I have no idea how anyone that reads that question could sit there and think of what you wrote out, in under two minutes. It's a great solution, but I think there must be some other way to crack this nut.


Yes, there is. You just read the question and the answer will be there by the time you are done (doing everything Bunuel did above).

Consider the question one line at a time:

"When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4."
Think - You will get x by finding the first such value which when divided by 5 leaves remainder 3 and when divided by 7 leaves remainder 4.
x will be 35N + First such value


"When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4."
Go back and compare the divisors and remainders. You see that they are exactly the same. This means y also belongs to the same series.
y = 35M + First such value

So you know now that x will be some multiple of 35 plus a number and y will be another multiple of 35 plus the same number. So x and y will differ by some multiple of 35.

"If x > y, which of the following must be a factor of x - y?"
So 35 must be a factor of x-y.

Answer (E)
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Manager
Manager
avatar
S
Joined: 17 Aug 2015
Posts: 92
GMAT ToolKit User
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 20 Jun 2016, 20:10
2
There is one other way to solve this problem. I earlier went with enumerating choices and it was taking time. So here is another way
x>y is given
x=5a1+3 and y=5a2+3..
also x=7b1+4 and y = 7b2+4
x-y = 5a1-5a2+3+3=> 5(a1-a2)
also x-y => 7b1-7b2+4-4=> 7(b1-b2).
so x-y is a multiple of both 5 and 7. 35 fits the bill it is a factor of 7 and 5.
Target Test Prep Representative
User avatar
V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 10547
Location: United States (CA)
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 06 Oct 2017, 09:54
shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?

A. 12
B. 15
C. 20
D. 28
E. 35


Let’s list some integers that have a remainder of 3 when divided by 5:

3, 8, 13, 18, 23, …

Let’s list some integers that have a remainder of 4 when divided by 7:

4, 11, 18, 25, …

We see that 18 is common in both lists and is the smallest positive integer that satisfies both conditions. Thus, y could be 18. To get a value for x, we can simply add the LCM of 5 and 7, i.e, 35, to 18. So, x could be 53 (notice that 53 also satisfies both conditions). We see that in this case, x - y = 35, and of all the choices, only 35 is factor of x - y.

Alternate Solution:

Since x produces a remainder of 3 when divided by 5, we can write x = 5p + 3 for some integer p.

Since x produces a remainder of 4 when divided by 7, we can write x = 7q + 4 for some integer q.

Since y produces a remainder of 3 when divided by 5, we can write y = 5s + 3 for some integer s.

Since y produces a remainder of 4 when divided by 7, we can write y = 7r + 4 for some integer r.

If we subtract the third equality from the first, we obtain: x - y = 5p - 5s = 5(p - s). Thus, x - y is a multiple of 5.

If we subtract the fourth equality from the second, we obtain: x - y = 7q - 7r = 7(q - r). Thus, x - y is a multiple of 7.

Since x - y is a multiple of both 5 and 7, it is a multiple of the LCM of 5 and 7 as well, namely 35.

Answer: E
_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
TTP - Target Test Prep Logo
202 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

VP
VP
avatar
D
Joined: 07 Dec 2014
Posts: 1252
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 01 Nov 2018, 07:52
shopaholic wrote:
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?

A. 12
B. 15
C. 20
D. 28
E. 35


because the divisors and remainders are the same for both dividends,
all values of x-y will equal a multiple of the product of the two divisors, 5 and 7
5*7=35
E
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 14970
Re: When positive integer x is divided by 5, the remainder is 3  [#permalink]

Show Tags

New post 02 Feb 2020, 13:43
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Bot
Re: When positive integer x is divided by 5, the remainder is 3   [#permalink] 02 Feb 2020, 13:43

When positive integer x is divided by 5, the remainder is 3

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne