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by Brent@GMATPrepNow » Wed Mar 12, 2014 8:01 am
sgr21 wrote:how many such numbers are there between 1& 500 that are multiples of either 3 or 7 or both?
NOTE: If you say BETWEEN 1 and 500, I'm assuming that we're NOT INCLUDING 1 and 500, otherwise we should say "from 1 to 500 INCLUSIVE"
Also, I'm assuming that you meant to say, "multiples of either 3 or 7 or both["

Okay, onto to solution...

Here's a nice rule: If x and y are multiples of k, then the number of multiples of k from x to y inclusive = [(y-x)/k] + 1
So, for example, the number of multiples of 3 from 6 to 21 inclusive = [(21 - 6)/3] + 1 = [15/3] + 1 = 6

For this question, we must recognize that: multiples of 3 or 7 or both = (multiples of 3) + (multiples of 7) - (multiples of 3 AND 7)
In other words, multiples of 3 or 7 or both = (multiples of 3) + (multiples of 7) - (multiples of 21)

multiples of 3
We want multiples of 3 from 3 to 498 inclusive
Using our formula, this = [(498 - 3)/3] + 1
= [495/3] + 1
= 166

multiples of 7
We want multiples of 7 from 7 to 497 inclusive
Using our formula, this = [(497 - 7)/7] + 1
= [490/7] + 1
= 71

multiples of 21
We want multiples of 21 from 21 to 483 inclusive
Using our formula, this = [(483 - 21)/21] + 1
= [462/21] + 1
= 23

So, multiples of 3 or 7 or both = 166 + 71 - 23) = 214

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Wed Mar 12, 2014 9:42 pm, edited 1 time in total.
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by Abhishek009 » Wed Mar 12, 2014 9:48 am
sgr21 wrote:how many such numbers are there between 1& 500 that are multiples of either 3 & 7 or both?
Form Pattern and sets -

Numbers which are multiples of 3 = { 3 , 6 , 9 , 12 , 15 , 18 , 21 ....... }

Total no of Numbers between 1 - 500 which are multiples of 3 is -

500/3 => 166

Numbers which are multiples of 7 = { 7 , 14 , 21 , 28 , 35 , 42 ...... }

Total no of Numbers between 1 - 500 which are multiples of 7 is -

500/7 => 71

Numbers which are multiples of both 3 as well as 7 = { 21 , 42 , 63.......... }

Total no of Numbers between 1 - 500 which are multiples of both 3 and 7 is -

500/21 => 23


Note that the multiples of 3 and 7 is common to both the sets 3 as well as 7 so we are actually double counting it and we need to subtract it to arrive at the correct answer.

So Total No of Numbers is -

166 + 71 - 23

=>214

What's the OA ?
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by GMATGuruNY » Wed Mar 12, 2014 10:46 am
sgr21 wrote:How many integers between 1 and 500, inclusive, are multiples of 3, 7 or both?
This is an OVERLAPPING GROUPS problem.

Total options = Multiples of 3 + Multiples of 7 - Multiples of 21.

The big idea is to SUBTRACT THE OVERLAP.
When we count the multiples of 3 and the multiples of 7, the OVERLAP between the two groups -- the multiples of 21 -- is counted twice.
Hence, the overlap -- the multiples of 21 -- must be subtracted from the total, as shown in the equation above.

Multiples of 3 between 1 and 500, inclusive:
Since 3 is not a factor of either endpoint (1 or 500), we can simply divide 3 into the total number of integers:
500/3 ≈ 166.

Multiples of 7 between 1 and 500, inclusive:
Since 7 is not a factor of either endpoint (1 or 500), we can simply divide 7 into the total number of integers:
500/7 ≈ 71.

Multiples of 21 between 1 and 500, inclusive:
Since 21 is not a factor of either endpoint (1 or 500), we can simply divide 21 into the total number of integers:
500/21 ≈ 23.

Thus:
Total options = 166 + 71 - 23 = 214.

Alternate approach:

Any integer that is a multiple of both 3 and 7 is a multiple of 21.
Count the number of integers between 1 and 21, inclusive, that are divisible by 3, 7, or both:
3, 6, 7, 9, 12, 14, 15, 18, 21
9 options.

Implication:
Between 1 and 500, inclusive, 9 of every 21 integers will be a multiple of 3, 7, or both:
9/21 * 500 = 3/7 * 500 ≈ 3 * 71.4 ≈ 214.2.
Thus, between 1 and 500, inclusive, the number of integers divisible by 3, 7 or both = 214.
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by theCodeToGMAT » Wed Mar 12, 2014 8:35 pm
Brent@GMATPrepNow wrote:
sgr21 wrote:how many such numbers are there between 1& 500 that are multiples of either 3 or 7 or both?
NOTE: If you say BETWEEN 1 and 500, I'm assuming that we're NOT INCLUDING 1 and 500, otherwise we should say "from 1 to 500 INCLUSIVE"
Also, I'm assuming that you meant to say, "multiples of either 3 or 7 or both["

Okay, onto to solution...

Here's a nice rule: If x and y are multiples of k, then the number of multiples of k from x to y inclusive = [(y-x)/k] + 1
So, for example, the number of multiples of 3 from 6 to 21 inclusive = [(21 - 6)/3] + 1 = [15/3] + 1 = 6

For this question, we must recognize that: multiples of 3 or 7 or both = (multiples of 3) + (multiples of 5) - (multiples of 3 AND 5)
In other words, multiples of 3 or 7 or both = (multiples of 3) + (multiples of 5) - (multiples of 15)

multiples of 3
We want multiples of 3 from 3 to 498 inclusive
Using our formula, this = [(498 - 3)/3] + 1
= [495/3] + 1
= 166

multiples of 5
We want multiples of 5 from 5 to 495 inclusive
Using our formula, this = [(495 - 5)/5] + 1
= [490/5] + 1
= 99

multiples of 15
We want multiples of 15 from 15 to 495 inclusive
Using our formula, this = [(495 - 15)/15] + 1
= [480/15] + 1
= 33

So, multiples of 3 or 7 or both = 166 + 99 - 33) = 232

Cheers,
Brent
Brent, you mistakenly used "5" for the second part; it's 7 :)
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by Brent@GMATPrepNow » Wed Mar 12, 2014 9:44 pm
theCodeToGMAT wrote: Brent, you mistakenly used "5" for the second part; it's 7 :)
You're absolutely right. Thanks Rahul!
I've edited my response accordingly.

Cheers,
Brent
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