Author 
Message 
TAGS:

Hide Tags

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7955
Location: Pune, India

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
03 May 2015, 20:27
vikasbansal227 wrote: Dear All,
Its difficult that on what basis we took "First common integer in two patterns" as a remainder for the formula?
Can someone please explain,
Thanks You want a number, n, that satisfies two conditions: "leaves a remainder of 4 after division by 6" and "a remainder of 2 after division by 8" So you find the first such number by writing down the numbers. You get that it is 10. 10 satisfies both conditions. What will be the next number? Any number that is a multiple of 6 more than 10 will continue to satisfy the first condition. e.g. 16, 22, 28, 34 etc Any number that is a multiple of 8 more than 10 will continue to satisfy the second condition. e.g. 18, 26, 34 etc So any number that is a multiple of both 6 and 8 more than 10 will satisfy both conditions. LCM of 6 and 8 is 24. 24 is divisible by both 6 and 8. So any number that is a multiple of 24 more than 10 will satisfy both conditions. e.g. 34, 58 .. etc These numbers can be written as 10 + 24a. Also, I think you did not check out the 4 links I gave here: positiveintegernleavesaremainderof4afterdivisionby9375240.html#p1496547They are very useful in understanding divisibility fundamentals.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Director
Joined: 10 Mar 2013
Posts: 584
Location: Germany
Concentration: Finance, Entrepreneurship
GPA: 3.88
WE: Information Technology (Consulting)

Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
27 Aug 2015, 00:43
Bunuel wrote: bchekuri wrote: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30? (A) 3 (B) 12 (C) 18 (D) 22 (E) 28
How to approach this Problem? Positive integer n leaves a remainder of 4 after division by 6 > \(n=6p+4\) > 4, 10, 16, 22, 28, ... Positive integer n leaves a remainder of 3 after division by 5 > \(n=5q+3\) > 3, 8, 13, 18, 23, 28, ... \(n=30k+28\)  we have 30 as lcm of 5 and 6 is 30 and we have 28 as the first common integer in the above patterns is 28. Hence remainder when positive integer n is divided by 30 is 28. Answer: E. P.S. n>30 is a redundant information. Hi Bunuel, do you think this is a valid approach for this type of questions in general ? I prefer algebra to number picking.Question 1n=6p+4, n=5q+3 > 6p+4=5q+3 > 6p+1=5q > 5p +(p+1)=5q, so p+1 must be a multiple of 5, means P=4,9,14 etc... if p=4: 6p+4=28 > q=5:5*5+3=28 => 28 is the first common number 28/30=0+28(R) Answer (E) Question 2n=6p+4, n=8q+2 /divide by 2 both expressions => 3q+(q1)=3p q1 must be a multiple of 3: 4,10,16 etc... let's pick 4 for q 8*4+2=34 > 34/12=2*12+10(R)
_________________
When you’re up, your friends know who you are. When you’re down, you know who your friends are.
Share some Kudos, if my posts help you. Thank you !
800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50 GMAT PREP 670 MGMAT CAT 630 KAPLAN CAT 660



Current Student
Joined: 11 Oct 2013
Posts: 120
Concentration: Marketing, General Management

Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
14 Dec 2015, 04:31
Bunuel wrote: To elaborate more.
Suppose we are told that: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?
The statement "positive integer n leaves a remainder of 4 after division by 6" can be expressed as: \(n=6p+4\). Thus according to this particular statement \(n\) could take the following values: 4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, ... The statement "positive integer n leaves a remainder of 2 after division by 8" can be expressed as: \(n=8q+2\). Thus according to this particular statement \(n\) could take the following values: 2, 10, 18, 26, 34, 42, 50, 58, 66, ...
The above two statements are both true, which means that the only valid values of \(n\) are the values which are common in both patterns. For example \(n\) can not be 16 (from first pattern) as the second formula does not give us 16 for any value of integer \(q\).
So we should derive general formula (based on both statements) that will give us only valid values of \(n\).
How can these two statement be expressed in one formula of a type \(n=kx+r\)? Where \(x\) is divisor and \(r\) is a remainder.
Divisor \(x\) would be the least common multiple of above two divisors 6 and 8, hence \(x=24\).
Remainder \(r\) would be the first common integer in above two patterns, hence \(r=10\).
Therefore general formula based on both statements is \(n=24k+10\). Thus according to this general formula valid values of \(n\) are: 10, 34, 58, ...
Now, \(n\) divided by 12 will give us the reminder of 10 (as 24k is divisible by 12).
Hope it helps. Thanks for the great explanation Bunuel. I have a doubt though. This seems to be valid when the final divisor mentioned in the question is a factor of LCM of the previous divisors. Are there any cases, where this would not be true? As in what if the question asked for divisibility by 14? i.e remainder of 24k+10 when divided by 14?
_________________
Its not over..



Intern
Joined: 21 Dec 2012
Posts: 4

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
09 Apr 2016, 06:26
How do we solve a question which has different remainder ? because in this case remainder is 2 , ie common to both the cases? What happens to the case when both the case have different remainder ? muralimba wrote: Friends,
IT IS VERY COMMON IN GMAT to solve this kind of qtns using "NEGATIVE REMAINDER" theory.
The theory says:
if a # x is devided by y and leave the positive # r as the remainder then it can also leave negative # (ry) as the remainder.
e.g:
9 when devided by 5 leves the remainder 4 : 9=5*1+4 it can also leave the remainder 45 = 1 : 9=5*2 1
back to the original qtn: n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5 ==> n leaves a remainder of 2 (i.e. 46) after division by 6 and a remainder of 2 (i.e. 35) after division by 5 ==> n when devided by 5 and 6 leaves the same remainder 2. what is n? LCM (5,6)2 = 302 = 28 CHECK: 28 when devided by 6 leaves the remainder 4 and when devided by 5 leaves the remainder 3
However, the qtn says n > 30
so what is the nex #, > 28, that can give the said remainders when devided by 6 and 5 nothing but 28 + (some multiple of 6 and 5) as this "some multiple of 6 and 5" will not give any remainder when devided by 5 or 6 but 28 will give the required remainders.
hence n could be anything that is in the form 28 + (some multiple of 6 and 5) observe that "some multiple of 6 and 5" is always a multiple of 30 as LCM (5,6) = 30.
hence when n (i.e. 28 + some multiple of 6 and 5) is devided by 30 gives the remainder 28.
ANSWER "E"
Regards, Murali.
Kudos?



Director
Joined: 07 Dec 2014
Posts: 907

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
31 Aug 2016, 19:11
r1=4 d1=6 r2=3 d2=5 n=r1+d1(d21) n=4+6(4)=28 next higher value of n=28+d1*d2=58 58/30 leaves a remainder of 28



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7955
Location: Pune, India

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
31 Aug 2016, 20:07
happy19 wrote: How do we solve a question which has different remainder ? because in this case remainder is 2 , ie common to both the cases? What happens to the case when both the case have different remainder ?
Check out this post. It discusses how to handle questions in which the remainders are different. http://www.veritasprep.com/blog/2011/05 ... spartii/
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Intern
Joined: 01 Sep 2016
Posts: 9

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
05 Oct 2016, 11:23
n = 6d + 4 n = 5d + 3
if we multiply the first equation by 5 and the second by 6 we get
5n = 30d + 20 (1) 6n = 30d + 18 (2)
if we make (1)  (2)
n = 2; n = 2
so 2/30 has a remainder of 28



Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98

Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
03 Dec 2016, 06:59
1
This post was BOOKMARKED
bchekuri wrote: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?
A. 3 B. 12 C. 18 D. 22 E. 28 \(N=4\) (mod 6)\(N= 3\) (mod 5)The difference between divisor and remainder is the same for both linear congruences \(6 4 = 5  3 = 2 = D\) Hence our N can b represented as: N = LCM (6, 5)*X  D, where X is >=0 and D is our common differnce. \(N = 30X  2\) X=0 > N=28 and for N>30 \(\frac{30*X  2}{30} = \frac{0  2}{30}\) = \(2\) (mod 30) = \(28\) (mod 30)Answer E.



Manager
Joined: 19 Aug 2016
Posts: 150
Location: India
GPA: 3.82

Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
16 Jan 2017, 12:08
Bunuel wrote: To elaborate more.
Suppose we are told that: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?
The statement "positive integer n leaves a remainder of 4 after division by 6" can be expressed as: \(n=6p+4\). Thus according to this particular statement \(n\) could take the following values: 4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, ... The statement "positive integer n leaves a remainder of 2 after division by 8" can be expressed as: \(n=8q+2\). Thus according to this particular statement \(n\) could take the following values: 2, 10, 18, 26, 34, 42, 50, 58, 66, ...
The above two statements are both true, which means that the only valid values of \(n\) are the values which are common in both patterns. For example \(n\) can not be 16 (from first pattern) as the second formula does not give us 16 for any value of integer \(q\).
So we should derive general formula (based on both statements) that will give us only valid values of \(n\).
How can these two statement be expressed in one formula of a type \(n=kx+r\)? Where \(x\) is divisor and \(r\) is a remainder.
Divisor \(x\) would be the least common multiple of above two divisors 6 and 8, hence \(x=24\).
Remainder \(r\) would be the first common integer in above two patterns, hence \(r=10\).
Therefore general formula based on both statements is \(n=24k+10\). Thus according to this general formula valid values of \(n\) are: 10, 34, 58, ...
Now, \(n\) divided by 12 will give us the reminder of 10 (as 24k is divisible by 12).
Hope it helps. Thanks for the amazing concept. I just have one more query: How will we apply this approach for questions such as: What is the remainder when the positive integer n is divided by 12? 1. When n is divided by 6, the remainder is 1. 2. When n is divided by 12, the remainder is greater than 5.
_________________
Consider giving me Kudos if you find my posts useful, challenging and helpful!



Math Expert
Joined: 02 Sep 2009
Posts: 43892

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
16 Jan 2017, 23:54
ashikaverma13 wrote: Bunuel wrote: To elaborate more.
Suppose we are told that: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?
The statement "positive integer n leaves a remainder of 4 after division by 6" can be expressed as: \(n=6p+4\). Thus according to this particular statement \(n\) could take the following values: 4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, ... The statement "positive integer n leaves a remainder of 2 after division by 8" can be expressed as: \(n=8q+2\). Thus according to this particular statement \(n\) could take the following values: 2, 10, 18, 26, 34, 42, 50, 58, 66, ...
The above two statements are both true, which means that the only valid values of \(n\) are the values which are common in both patterns. For example \(n\) can not be 16 (from first pattern) as the second formula does not give us 16 for any value of integer \(q\).
So we should derive general formula (based on both statements) that will give us only valid values of \(n\).
How can these two statement be expressed in one formula of a type \(n=kx+r\)? Where \(x\) is divisor and \(r\) is a remainder.
Divisor \(x\) would be the least common multiple of above two divisors 6 and 8, hence \(x=24\).
Remainder \(r\) would be the first common integer in above two patterns, hence \(r=10\).
Therefore general formula based on both statements is \(n=24k+10\). Thus according to this general formula valid values of \(n\) are: 10, 34, 58, ...
Now, \(n\) divided by 12 will give us the reminder of 10 (as 24k is divisible by 12).
Hope it helps. Thanks for the amazing concept. I just have one more query: How will we apply this approach for questions such as: What is the remainder when the positive integer n is divided by 12? 1. When n is divided by 6, the remainder is 1. 2. When n is divided by 12, the remainder is greater than 5. This question is discussed here: whatistheremainderwhenthepositiveintegernisdivided161170.html
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Joined: 05 Dec 2016
Posts: 263
Concentration: Strategy, Finance

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
27 Feb 2017, 05:45
6k+4 => 6k2 5q+3 => 5q2 Hence, 30m2 => 2+ 30 = 28



Manager
Joined: 27 Dec 2016
Posts: 152
Concentration: International Business, Marketing

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
03 Jun 2017, 15:51
Bunuel wrote: bchekuri wrote: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30? (A) 3 (B) 12 (C) 18 (D) 22 (E) 28
How to approach this Problem? Positive integer n leaves a remainder of 4 after division by 6 > \(n=6p+4\) > 4, 10, 16, 22, 28, ... Positive integer n leaves a remainder of 3 after division by 5 > \(n=5q+3\) > 3, 8, 13, 18, 23, 28, ... \(n=30k+28\)  we have 30 as lcm of 5 and 6 is 30 and we have 28 as the first common integer in the above patterns is 28. Hence remainder when positive integer n is divided by 30 is 28. Answer: E. P.S. n>30 is a redundant information. Hi Bunuel, I understood everything up to this point "Hence remainder when positive integer n is divided by 30 is 28" I was wondering could you please explain how you got 28 from this statement? Would greatly appreciate it!! Thank You!



Director
Joined: 07 Dec 2014
Posts: 907

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
03 Jun 2017, 19:58
bchekuri wrote: Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?
A. 3 B. 12 C. 18 D. 22 E. 28 assume difference of 1 between n/6 and n/5 quotients (n3)/5(n4)/6=1 n=28 28+5*6=58 58/30 gives a remainder of 28 E



Intern
Joined: 30 Aug 2017
Posts: 22
Concentration: Real Estate, Operations

Re: Positive integer n leaves a remainder of 4 after division by [#permalink]
Show Tags
11 Sep 2017, 00:38
I think one easy way to solve this is as follows: Given: n=6q+4 and n=5p+3 multiply first eq by 5 and 2nd by 6(in order to get 30 as factor in quotient) therefore we have: 5n=30q+20 (a) 6n=30p+18 (b)
ba: n=30(pq)2 since remainder is 2, positive remainder =302=28 (option E)




Re: Positive integer n leaves a remainder of 4 after division by
[#permalink]
11 Sep 2017, 00:38



Go to page
Previous
1 2 3
[ 54 posts ]



