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Manager  Joined: 29 Jun 2010
Posts: 225
Schools: LBS, Oxford
How many positive integers less than 5,000 are evenly divisi  [#permalink]

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18 00:00

Difficulty:   95% (hard)

Question Stats: 61% (03:06) correct 39% (03:02) wrong based on 381 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
Manager  Joined: 14 Mar 2011
Posts: 184
Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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5
5
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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SVP  Joined: 06 Sep 2013
Posts: 1569
Concentration: Finance
Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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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 V
Joined: 02 Sep 2009
Posts: 58445
Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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1
3
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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Manager  Joined: 20 Dec 2013
Posts: 224
Location: India
Re: Integers less than 5,000 are not divisible by 15 or 21 ?  [#permalink]

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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!
Manager  Joined: 20 Dec 2013
Posts: 224
Location: India
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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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 $$answer=4999-333-238+47=4475$$
Intern  Joined: 26 Mar 2014
Posts: 4
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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2
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!
Manager  Joined: 26 May 2013
Posts: 52
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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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.
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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Amit0507 wrote:
hey Bunuel,
What does evenly divisible number means? I thought the quotient for such division should be multiple of 2.

Thanks,
Amit

Nope.

Evenly divisible means divisible without a remainder, so simply divisible.
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Joined: 27 Aug 2014
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GMAT 1: 660 Q45 V35 GPA: 3.66
WE: Consulting (Consulting)
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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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 ?
Intern  B
Joined: 01 Mar 2017
Posts: 14
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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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 V
Joined: 02 Sep 2009
Posts: 58445
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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1
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, ...
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Intern  Joined: 18 Dec 2018
Posts: 46
Re: How many positive integers less than 5,000 are evenly divisi  [#permalink]

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We can find the no of integers divisible by 15 using the formula of A.P. with first term as 15 and common difference also 15.
Since, the largest term smaller than 5000 divisible by 15 is 4995.
4995 = 15 + (n-1)15. We get, n = 333.
Similarly, we can find no of terms divisible by 21.
So, no of integers from 0 to 5000 divisible by 21 = 238.
Now, we need to find the no of integers divisible by 15 x 21 because these integers are counted twice as the factors of 15 as well as the factors of 21.
No of factors of LCM of 15 &21 till 5000 = 47.
So, total no of positive integers till 5000 divisible by either 15 or 21 = 333 + 238 – 47 = 524.
Hence, no of integers not divisible by 15 or 21 = 4999 – 524 = 4475. Re: How many positive integers less than 5,000 are evenly divisi   [#permalink] 18 Dec 2018, 23:30
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