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How many positive integers less than 5,000 are evenly divisi

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How many positive integers less than 5,000 are evenly divisi  [#permalink]

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New post 12 Jul 2011, 22:20
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Question Stats:

63% (03:04) correct 37% (03:05) wrong based on 353 sessions

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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
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Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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New post 12 Jul 2011, 23:00
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4
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.
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Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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New post 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
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Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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New post 04 Apr 2014, 08:12
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2
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:
what-is-the-number-of-integers-from-1-to-1000-inclusive-126153.html
what-is-the-total-number-of-positive-integers-that-are-less-128104.html
how-many-integers-from-1-to-200-inclusive-are-divisib-109333.html
how-many-even-integers-n-where-100-n-200-are-divisib-103779.html

Hope it helps.
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Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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New post 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:
what-is-the-number-of-integers-from-1-to-1000-inclusive-126153.html
what-is-the-total-number-of-positive-integers-that-are-less-128104.html
how-many-integers-from-1-to-200-inclusive-are-divisib-109333.html
how-many-even-integers-n-where-100-n-200-are-divisib-103779.html

Hope it helps.


Hi Bunuel,
Is there a shorter way around this question?This approach took me 5 minutes!
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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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New post 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 \(answer=4999-333-238+47=4475\)
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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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New post 16 Apr 2014, 23:53
1
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 \(answer=4999-333-238+47=4475\)


This seems to be one of those killer problems that aren't necessarily conceptually difficult, but unless you have (what for me is) very high-powered 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 20-30 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!
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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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New post 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:
what-is-the-number-of-integers-from-1-to-1000-inclusive-126153.html
what-is-the-total-number-of-positive-integers-that-are-less-128104.html
how-many-integers-from-1-to-200-inclusive-are-divisib-109333.html
how-many-even-integers-n-where-100-n-200-are-divisib-103779.html

Hope it helps.
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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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

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New post 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/what-is-the-n ... 26153.html
http://gmatclub.com/forum/what-is-the-t ... 28104.html
http://gmatclub.com/forum/how-many-inte ... 09333.html
http://gmatclub.com/forum/how-many-even ... 03779.html

Hope it helps.


In questions like these do we always have to take an LCM ?
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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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New post 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?

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Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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