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

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When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 02 Mar 2012, 09:39
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Jun 2013, 05:31, edited 2 times in total.
Edited the question, added the answer choices and OA
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Re: How to solve this problem [#permalink] New post 02 Mar 2012, 10:16
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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.
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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 06 Jun 2013, 05:32
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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

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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 06 Jun 2013, 08:50
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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
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Re: How to solve this problem [#permalink] 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.
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Re: How to solve this problem [#permalink] New post 23 Oct 2013, 23:29
Expert's post
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.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] 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?
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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 24 Oct 2013, 08:04
Expert's post
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.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 25 Nov 2014, 15:36
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Re: When positive integer x is divided by 5, the remainder is 3 [#permalink] New post 25 Nov 2014, 20:30
Expert's post
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)
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Re: When positive integer x is divided by 5, the remainder is 3   [#permalink] 25 Nov 2014, 20:30
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