Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 29 Jun 2010
Posts: 230
Schools: LBS, Oxford

How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
12 Jul 2011, 22:20
Question Stats:
63% (03:04) correct 37% (03:05) wrong based on 353 sessions
HideShow timer Statistics
How many positive integers less than 5,000 are evenly divisible by neither 15 nor 21? A. 4,514 B. 4,475 C. 4,521 D. 4,428 E. 4,349
Official Answer and Stats are available only to registered users. Register/ Login.




Manager
Joined: 14 Mar 2011
Posts: 193

Re: Integers less than 5,000 are not divisible by 15 or 21 ?
[#permalink]
Show Tags
12 Jul 2011, 23:00
integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B.




SVP
Joined: 06 Sep 2013
Posts: 1765
Concentration: Finance

Re: Integers less than 5,000 are not divisible by 15 or 21 ?
[#permalink]
Show Tags
04 Apr 2014, 06:10
Aj85 wrote: integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B. Question here, why don't you multiply 47*2 I thought that 105 was contained in both 15 and 21? Is there any reason why you don't do this? Thanks Cheers J



Math Expert
Joined: 02 Sep 2009
Posts: 50058

Re: Integers less than 5,000 are not divisible by 15 or 21 ?
[#permalink]
Show Tags
04 Apr 2014, 08:12
jlgdr wrote: Aj85 wrote: integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B. Question here, why don't you multiply 47*2 I thought that 105 was contained in both 15 and 21? Is there any reason why you don't do this? Thanks Cheers J 47 numbers are counted TWICE, we need them to be counted ONCE, so that's why are subtracting 47 once. Similar questions to practice: whatisthenumberofintegersfrom1to1000inclusive126153.htmlwhatisthetotalnumberofpositiveintegersthatareless128104.htmlhowmanyintegersfrom1to200inclusivearedivisib109333.htmlhowmanyevenintegersnwhere100n200aredivisib103779.htmlHope it helps.
_________________
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



Manager
Joined: 20 Dec 2013
Posts: 238
Location: India

Re: Integers less than 5,000 are not divisible by 15 or 21 ?
[#permalink]
Show Tags
04 Apr 2014, 21:57
Bunuel wrote: jlgdr wrote: Aj85 wrote: integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B. Question here, why don't you multiply 47*2 I thought that 105 was contained in both 15 and 21? Is there any reason why you don't do this? Thanks Cheers J 47 numbers are counted TWICE, we need them to be counted ONCE, so that's why are subtracting 47 once. Similar questions to practice: whatisthenumberofintegersfrom1to1000inclusive126153.htmlwhatisthetotalnumberofpositiveintegersthatareless128104.htmlhowmanyintegersfrom1to200inclusivearedivisib109333.htmlhowmanyevenintegersnwhere100n200aredivisib103779.htmlHope it helps. Hi Bunuel, Is there a shorter way around this question?This approach took me 5 minutes!



Manager
Joined: 20 Dec 2013
Posts: 238
Location: India

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
04 Apr 2014, 21:59
Option B.(Took 5 minutes to solve with the following approach!) All nos. from \(1\) to \(5000\) since we're given POSITIVE integers.\(=4999\) Multiples of \(15\) in this range\(=4995/15=333\) Multiples of \(21=4998/21=238\) Multiples of both \(15\) and \(21\)=multiples of \(105=47\) Now \(answer=4999333238+47=4475\)



Intern
Joined: 26 Mar 2014
Posts: 4

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
16 Apr 2014, 23:53
AKG1593 wrote: Option B.(Took 5 minutes to solve with the following approach!) All nos. from \(1\) to \(5000\) since we're given POSITIVE integers.\(=4999\) Multiples of \(15\) in this range\(=4995/15=333\) Multiples of \(21=4998/21=238\) Multiples of both \(15\) and \(21\)=multiples of \(105=47\) Now \(answer=4999333238+47=4475\) This seems to be one of those killer problems that aren't necessarily conceptually difficult, but unless you have (what for me is) very highpowered arithmetic (long division) skills you're going to be in a time crunch here and in danger of compromising your quant section on test day. Practice, practice, practice those fundamentals! One trick, because the ACs are so close together, is that you want to be careful to account for the fact that "less than 5000" accounts for only 4,999 integers, not including 5000. (Turns out on this problem it's not of such great concern, but on many other problems, it definitely is!) For this question, huge waste of about 2030 seconds was the initial rephrase: "How many integers.. divisible by NEITHER 15 or 21." Don't know why but that tripped me up. Do they mean all of the numbers not divisible by 105 (Neither 15 AND 21, I saw the language trick there and ruled this out but not after wasting some time on math). What were are therefore looking for is Opposite: 4999 (minus) all the numbers that 15 or 21 go into (minus) overlap between multiples of 15 and 21. Let the long division begin!



Manager
Joined: 26 May 2013
Posts: 52

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
21 Nov 2014, 19:11
hey Bunuel, What does evenly divisible number means? I thought the quotient for such division should be multiple of 2. Thanks, Amit Bunuel wrote: jlgdr wrote: Aj85 wrote: integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B. Question here, why don't you multiply 47*2 I thought that 105 was contained in both 15 and 21? Is there any reason why you don't do this? Thanks Cheers J 47 numbers are counted TWICE, we need them to be counted ONCE, so that's why are subtracting 47 once. Similar questions to practice: whatisthenumberofintegersfrom1to1000inclusive126153.htmlwhatisthetotalnumberofpositiveintegersthatareless128104.htmlhowmanyintegersfrom1to200inclusivearedivisib109333.htmlhowmanyevenintegersnwhere100n200aredivisib103779.htmlHope it helps.



Math Expert
Joined: 02 Sep 2009
Posts: 50058

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
22 Nov 2014, 06:36



Manager
Joined: 27 Aug 2014
Posts: 56
Location: Canada
Concentration: Strategy, Technology
GPA: 3.66
WE: Consulting (Consulting)

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
21 Feb 2017, 13:49
Bunuel wrote: jlgdr wrote: Aj85 wrote: integers less than 5000 divisible by 15 5000/15 =333.something , so 333
integers less than 5000 divisible by 21 5000/21 = 238.## , so 238
we have double counted some, so take LCM of 15 and 21 =105 and divide by 5000, we get 47. so all numbers divisible by 15 and 21 = 333 +238 47 =524
now subtract that from 4999. 4999 524 = 4475 answer B. Question here, why don't you multiply 47*2 I thought that 105 was contained in both 15 and 21? Is there any reason why you don't do this? Thanks Cheers J 47 numbers are counted TWICE, we need them to be counted ONCE, so that's why are subtracting 47 once. Similar questions to practice: http://gmatclub.com/forum/whatisthen ... 26153.htmlhttp://gmatclub.com/forum/whatisthet ... 28104.htmlhttp://gmatclub.com/forum/howmanyinte ... 09333.htmlhttp://gmatclub.com/forum/howmanyeven ... 03779.htmlHope it helps. In questions like these do we always have to take an LCM ?



Intern
Joined: 01 Mar 2017
Posts: 14

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
03 Apr 2018, 00:38
I’m sorry for my ignorance, but why do we take the LCM of 15 and 21 to remove double count? I multiplied 15*21= 315 and divided 5000/315= 15 times to get the overlap. I know it is wrong now, but can someone explain it to me please?
Posted from my mobile device



Math Expert
Joined: 02 Sep 2009
Posts: 50058

Re: How many positive integers less than 5,000 are evenly divisi
[#permalink]
Show Tags
03 Apr 2018, 03:41
Juliaz wrote: I’m sorry for my ignorance, but why do we take the LCM of 15 and 21 to remove double count? I multiplied 15*21= 315 and divided 5000/315= 15 times to get the overlap. I know it is wrong now, but can someone explain it to me please?
Posted from my mobile device The numbers which are divisible by both 15 and 21 are numbers which are divisible by the least common multiple of 15 and 21, which is 105: 105, 210, 315, 420, ... If you simply multiply you are loosing many numbers, for example, 105, 210, 420, ...
_________________
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




Re: How many positive integers less than 5,000 are evenly divisi &nbs
[#permalink]
03 Apr 2018, 03:41






