GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 22 Oct 2018, 00:48

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 n is divided by 13, the remainder is 2. When n

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

Hide Tags

GMATH Teacher
User avatar
S
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 384
Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

Show Tags

New post 30 Sep 2018, 15:30
Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

\(1 \leqslant n \leqslant 179\,\,\,\,\,\left( {n\,\,\,\operatorname{int} } \right)\,\,\,\,\left( * \right)\)
\(n = 13M + 2\,\,\,\,,\,\,\,M\,\,\operatorname{int} \,\,\,\,\left( {\text{I}} \right)\)
\(n = 8J + 5\,\,\,\,,\,\,\,J\,\,\operatorname{int} \,\,\,\,\left( {{\text{II}}} \right)\)

\(?\,\,\,:\,\,\,n\,\,\,{\text{in}}\,\,\,\left( * \right) \cap \left( {\text{I}} \right) \cap \left( {{\text{II}}} \right)\)

\(\left( * \right)\,\, \cap \,\,\left( {\text{I}} \right)\,\,\,:\,\,\,\,\,\,1\,\, \leqslant \,\,13M + 2\,\, \leqslant \,\,179\,\,\,\,\,\mathop \Leftrightarrow \limits^{ - \,2} \,\,\,\,\, - 1 \leqslant 13M \leqslant 177\,\,\left( { = 169 + 8} \right)\)
\(- 1 \leqslant 13M \leqslant 177\,\,\left( { = 169 + 8} \right)\,\,\,\,\,\mathop \Leftrightarrow \limits^{M\,\,\operatorname{int} } \,\,\,0 \leqslant 13M \leqslant 169\,\,\,\,\,\mathop \Leftrightarrow \limits^{:\,\,13} \,\,\,\,0 \leqslant M \leqslant 13\)

\(\left( * \right)\,\, \cap \,\,\left( {{\text{II}}} \right)\,\,\,:\,\,\,\,\,\,\left\{ \begin{gathered}
\,n - 5\,\,{\text{divisible}}\,\,{\text{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,n - 5\,\,\,\,\,{\text{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,\,{\text{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {\text{I}} \right)} \,\,\,\,\,M\,\,{\text{odd}} \hfill \\
\,n - 5\mathop = \limits^{\left( {\text{I}} \right)} 13M - 3\,\,{\text{divisible}}\,\,{\text{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,\frac{{13M - 3}}{2}\,\,\,\,{\text{divisible}}\,\,{\text{by}}\,\,4\,\,\,\left( {***} \right)\,\,\,\,\,\, \hfill \\
\end{gathered} \right.\)

\(\left. \begin{gathered}
M = 1\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 - 3}}{2} = 5\,\,\,{\text{odd}}\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 3\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 3 - 3}}{2} = 18\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 5\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 5 - 3}}{2} = 31\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}}\,\,\,\,\,\, \hfill \\
\boxed{M = 7}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 7 - 3}}{2} = 44\,\,\,\,\left( {***} \right)\,\,\,{\text{YES}} \hfill \\
M = 9\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 11\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 11 - 3}}{2} = 70\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 13\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________

Fabio Skilnik :: https://www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 31/Oct with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 60% discount!

CEO
CEO
User avatar
P
Joined: 08 Jul 2010
Posts: 2564
Location: India
GMAT: INSIGHT
WE: Education (Education)
Reviews Badge
Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

Show Tags

New post 01 Oct 2018, 03:08
1
Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


Kudos for a correct solution.


13x+2 = 8y+5

Values of the format 13x+2 are 2, 15, 28, 41, 54, 67, 80, 93, 106...

Values of the format 8y+5 are 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93...

The first common Solution as highlighted is 93

every next solution will be at a gap that is multiple of 8 as well as a multiple of 13 i.e. at a gap that is LCM of 13 and 8 which is 104

Hence, Next solution = 93+104 = 197
Hence, Next solution = 197+104 = 301 etc.

But we have to find solutions less than 180 hence 93 is the only possible solution

Answer: Option B
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Senior Manager
Senior Manager
avatar
S
Joined: 04 Aug 2010
Posts: 296
Schools: Dartmouth College
Re: When positive integer n is divided by 13, the remainder is 2. When n  [#permalink]

Show Tags

New post 01 Oct 2018, 03:34
A quick lesson on remainders:

Quote:
When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.

When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.

Combined, the statements above imply that when x is divided by 35 -- the LOWEST COMMON MULTIPLE OF 5 AND 7 -- there will be a constant remainder R.
Put another way, x is R more than a multiple of 35:
x = 35c + R.

To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13, 18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11, 18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.

Putting it all together:
x = 35c + 18.

Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10, 13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2, 13...

Thus, when x is divided by 33 -- the LCM of 3 and 11 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...


Onto the problem at hand:

Bunuel wrote:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


When n is divided by 13, the remainder is 2.
n = 13a + 2 = 2, 15, 28, 41, 54, 67, 80, 93...
When n is divided by 8, the remainder is 5.
n = 8b + 5 = 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93...

Thus, when n is divided by 104 -- the LCM of 13 and 8 -- the remainder will be 93 (the smallest value common to both lists).
n = 104c + 93 = 93, 197...

In the resulting list, only the value in green is less than 180.


_________________

GMAT and GRE Tutor
Over 1800 followers
Click here to learn more
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.

GMAT Club Bot
Re: When positive integer n is divided by 13, the remainder is 2. When n &nbs [#permalink] 01 Oct 2018, 03:34

Go to page   Previous    1   2   [ 23 posts ] 

Display posts from previous: Sort by

When positive integer n is divided by 13, the remainder is 2. When n

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


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

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®.