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# What is the number of integers from 1 to 1000 (m07q14)

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What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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12 Feb 2009, 23:16
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What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

(A) 884
(B) 890
(C) 892
(D) 910
(E) 945

[Reveal] Spoiler: OA
A

Source: GMAT Club Tests - hardest GMAT questions

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Re: What is the number of integers from 1 to 1000... [#permalink]

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12 Feb 2009, 23:49
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What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

* 884
* 890
* 892
* 910
* 945
---------------------------------
We can go this way:

Calculate the no. of terms from 1 to 1000 (inclusive) that are divisible by 11 or 35 or both.

1.) Total no. of terms divisible by 11 are 90. We can calculate this by finding the first and last terms, which are 11 & 990 respectively. Then we will find the total no. of terms by using equation

Last Term = a + (n-1)d where a=11, d=11, Last Term=990.

So, n=90

2.) Similarly, total no. of terms divisible by 35 are 28. Find it using the above method.

3.) To find terms divisible by both 11 & 35, find the first term. Since both have no common factors except 1, just multiply 11 & 35 to get the first common term i.e., 385. Next term is 770.

So, in total, there are 2 common terms for 11 & 35.
------------------------------

Hence, the total no. of terms from 1 to 1000 (inclusive) that are divisible by 11 or 35 or both = 90 + 28 - 2 = 116

So, the correct answer = 1000 - 116 = 884, which will give us the total no. of terms that are divisible neither by 11 nor 35.

So, I'll go for first option, i.e., 884

Though the explanation looks a bit lengthy, it'll not take much time to solve.

HTH
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Last edited by Technext on 13 Feb 2009, 03:13, edited 3 times in total.

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Re: What is the number of integers from 1 to 1000... [#permalink]

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13 Feb 2009, 16:11
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xALIx wrote:
What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

* 884
* 890
* 892
* 910
* 945

1000/11 = 90.xx
divisible 11 = 90

1000/35 = 28.x
Divisible by 35 = 28

We need to exclude 11*35 and 2*11*35 numbers are counted twice.

Anser = 1000-(90+28-2) =1000-116=884
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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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09 Apr 2012, 22:19
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Hello,

Calculate the no. of terms from 1 to 1000 (inclusive) that are divisible by 11 or 35 or both.

1. No of terms divisible by 11 -> 1000/11 = 90
2. No of terms divisible by 35 -> 1000/35 = 28
3. No of terms divisible by 11 and 35 -> 1000/(11*35) = 2

Answer = 1000- (90+28-2) = 884.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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31 Mar 2010, 05:31
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Numbers Divisible by 11:
(1000/11) + (1000/(11^2)) = 90+8 = 98
Numbers Divisible by 35:
1000/35 = 20
Numbers Divisible by 385 (11*35):
1000/385 = 2

Hence, Numbers Divisible by both 11 and 35 = 98+20-2 = 116
Therefore, Numbers Not Divisible = 1000-116= 884

IMO A.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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09 Apr 2013, 05:22
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What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

(A) 884
(B) 890
(C) 892
(D) 910
(E) 945

1.From 1-1000 numbers divisible by 11 = 990/11 = 90
2.From 1-1000 numbers divisible by 35 = 1050/35 -2 = 28
3.Common numbers(div by 11 and 35) = 35 *11 and 35 *22 (2nos)
So numbers not div = 1000-(90+28-2) = 884

IMO A

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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09 Apr 2013, 06:07
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Number of numbers divisible by 11 = 1000/11 = 90
Number of numbers divisible by 35 = 1000/35 = 28

Number of numbers divisible by both 11 and 35 = 1000/(35*11) = 1000/385 = 2.

So,total number of numbers divisible by both 11 and 35 = 90 + 28 - 2 = 116.

Then,the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35 = 1000 - 116 = 884.

--------------------------------------------------

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Last edited by subhendu009 on 09 Apr 2013, 06:55, edited 1 time in total.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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06 Feb 2010, 10:18
we are subtracting 2 to avoid double counting ..( was initially breaking my head )
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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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31 Mar 2010, 07:34
deepak4mba wrote:
Numbers Divisible by 11:
(1000/11) + (1000/(11^2)) = 90+8 = 98
Numbers Divisible by 35:
1000/35 = 20
Numbers Divisible by 385 (11*35):
1000/385 = 2

Hence, Numbers Divisible by both 11 and 35 = 98+20-2 = 116
Therefore, Numbers Not Divisible = 1000-116= 884

IMO A.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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31 Mar 2010, 08:33
deepak4mba wrote:
Numbers Divisible by 11:
(1000/11) + (1000/(11^2)) = 90+8 = 98
Numbers Divisible by 35:
1000/35 = 20
Numbers Divisible by 385 (11*35):
1000/385 = 2

Hence, Numbers Divisible by both 11 and 35 = 98+20-2 = 116
Therefore, Numbers Not Divisible = 1000-116= 884

IMO A.

Deepak, 1000/35 is 28 not 20.. How did you get 20? Please explain .

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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04 Apr 2011, 06:11
1000/11 = 90 ( no of integers divisible by 11 and less than 1000)

1000/35 = 28 ( no of integers divisible by 11 and less than 35)

No of Common factors = 1 , 11*35
=2

answer =1000-( 90+28-2 ) = 884

A

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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04 Apr 2011, 18:10
So total multiples of 35 in 1000 = Quotient of 1000/35 = 28

For 11, total multiples = 1000/11 = 90

But there are common multiples as 385,770 which should be counted only once

Because - 35 and 11 LCM = 7 * 5 * 11 = 35 * 11 = 185 so 385 * 2 = 770

So total = 90 + 28 - 2 = 90 + 26 = 116

Hence not divisible = 1000 - 116 = 884

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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07 Apr 2011, 11:12
no.s div by 11 =90
no.s div by 35 =28
rem = 1000-(90+28) = 882
but 2 no.s div by both 11 and 35

hence, my ans: 882+2=884 i.e., A

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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06 Apr 2012, 20:25
xALIx wrote:
What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

(A) 884
(B) 890
(C) 892
(D) 910
(E) 945

[Reveal] Spoiler: OA
A

Source: GMAT Club Tests - hardest GMAT questions

My answer is 890, i think i have not calculated the common terms divisible by both 11 & 35... so might be lesser....
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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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07 Apr 2012, 03:03
xALIx wrote:
What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

(A) 884
(B) 890
(C) 892
(D) 910
(E) 945

[Reveal] Spoiler: OA
A

Source: GMAT Club Tests - hardest GMAT questions

# of multiples of 11 in the given range (last-first)/multiple+1=(990-11)/11+1=90 (check this: totally-basic-94862.html);
# of multiples of 35 in the given range (last-first)/multiple+1=(980-35)/35+1=28;
# of multiples of both 11 and 35 is 2 (11*35=385 and 770);

So, # of multiples of 11 or 35 in the given range is 90+28-2=116. Thus numbers which are not divisible by either of them is 1000-116=884.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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11 Apr 2012, 08:31
Hi,

How can you rephrase the question in problem solving method and also in Data sufficiency method.

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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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11 Apr 2012, 08:45
maryann wrote:
Hi,

How can you rephrase the question in problem solving method and also in Data sufficiency method.

Unfortunately your question is not clear at all. Also are you sure you've posted it in the right place?
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Re: What is the number of integers from 1 to 1000... [#permalink]

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09 Apr 2013, 06:28
x2suresh wrote:
xALIx wrote:
What is the number of integers from 1 to 1000 (inclusive) that are not divisible by 11 nor by 35?

* 884
* 890
* 892
* 910
* 945

1000/11 = 90.xx
divisible 11 = 90

1000/35 = 28.x
Divisible by 35 = 28

We need to exclude 11*35 and 2*11*35 numbers are counted twice.

Anser = 1000-(90+28-2) =1000-116=884

That's what was going on in my mind, but I missed out because of a lack of clarity in understanding "neither 11 nor 35" & double counted the common ones...

Thank you for completing the simple effective analysis!
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Re: What is the number of integers from 1 to 1000 (m07q14) [#permalink]

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09 Apr 2013, 08:48
In order to find the count of numbers between 1 and 1000 that are divisible by neither 11 nor 35, first find the count of numbers that are divisible by either of the numbers, then subtract that number from 1000 to get to the answer.

To find the count of the numbers that are divisible by either of the numbers, find the count of positive multiples of 11, that of 35, and that of 11*35 in the given range. Then add the first two counts; subtract the third count. Then, subtract this number from 1000 to get to the answer.

Count of multiple of 11 less than 1000:
11.x < 1000
=> x < 1000/11
=> x < 90.9
Therefore, the count of positive numbers less than 1000 and divisible by 11 is 90.

Count of multiple of 35 less than 1000:
35.y < 1000
=> y < 1000/35
=> y < 28.5
Therefore, the count of positive numbers less than 1000 and divisible by 35 is 28.

Count of multiple of 11*35 less than 1000:
11.35.z < 1000
=> z < 1000/35*11
=> z < 2.5
Therefore, the count of positive numbers less than 1000 and divisible by 11*35 is 2.

Count of the positive number less than 1000 and divisible by either of the numbers = 90 + 28 - 2 = 116

So, the count of the numbers between 1 and 1000 that are divisible by neither 11 nor 35 = 1000 - 116 = 884

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Re: What is the number of integers from 1 to 1000 (m07q14)   [#permalink] 09 Apr 2013, 08:48
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# What is the number of integers from 1 to 1000 (m07q14)

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