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How many positive integers less than 1000 are multiples of 5  [#permalink]

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61 00:00

Difficulty:   95% (hard)

Question Stats: 39% (02:57) correct 61% (02:52) wrong based on 493 sessions

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How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

Originally posted by kevincan on 14 Apr 2007, 22:34.
Last edited by Bunuel on 30 Apr 2015, 02:34, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Intern  Joined: 27 Feb 2007
Posts: 39
Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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11
15
It's 129 (D)

There are 199 multiples of 5 that are less than 1000.

Now we have to exclude the multiples of 4 and 7 from these 199 numbers.

Let's first find out the numbers below 1000 that are multiples of 5 as well as 4. (that is, multiples of 20). There are 49 such numbers.

Ditto for 5 as well as 7. There are 28 such numbers.

Subtracting these numbers from 199, we get, 199-49-28=122.

However, there will be some numbers which are multiples of both 4 and 7 which have been subtracted twice in the above calculation.

There are 7 such numbers (multiples of 20 as well as 35, that is multiples of 140).

Thus the required number is 122+7 = 129.
##### General Discussion
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114 (B) 121 (C) 122 (D) 129 (E) 136

There are 1000/5=200 multiples of 5 . now we have to find the number of multiples of 5 and 4 that is 20 between 0 to 1000 and 5 and 7 that is 35 between 0 to 1000.

multiples of 20=1000/20=50 multiples. and multiples of 35= 1000/35 =28.5 so 28 multiples of 35.

So multiples of 5 between o to 1000 not including multiples of 20 and 35 are equal to 200-78 = 122 multiples.

Javed.

Cheers!
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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Javed's method seems to be straight forward and correct.

Although in a counting problem such as this, B is tempting because it's very close to C and it might make you think that you forgot to subtract 1 from somewhere...

What is the OA?
Manager  Joined: 01 Jan 2007
Posts: 220
Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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rakesh.id wrote:
It's 129 (D)

There are 199 multiples of 5 that are less than 1000.

Now we have to exclude the multiples of 4 and 7 from these 199 numbers.

Let's first find out the numbers below 1000 that are multiples of 5 as well as 4. (that is, multiples of 20). There are 49 such numbers.

Ditto for 5 as well as 7. There are 28 such numbers.

Subtracting these numbers from 199, we get, 199-49-28=122.

However, there will be some numbers which are multiples of both 4 and 7 which have been subtracted twice in the above calculation.

There are 7 such numbers (multiples of 20 as well as 35, that is multiples of 140).

Thus the required number is 122+7 = 129.

Javed

Cheers!
Manager  Joined: 25 Mar 2007
Posts: 50
Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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1
multiples of 5: 5(1,2,3,4,5......, 199), 199 in total < 1000
multiples of 4 and 5: 20(1,2,3,4,5,....,49) 49 in total < 1000
multiples of 5 and 7: 35(1,2,3,4,5,....,28) 28 in total < 1000

199 - 49 - 28 = 122

The double counts include multiples of 20,35 (2*2*5, 7*5)
70,140,350,700,770,910,980

122 + 7(doubles) =129
Intern  Joined: 27 Feb 2007
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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alfyG wrote:
multiples of 5: 5(1,2,3,4,5......, 199), 199 in total < 1000
multiples of 4 and 5: 20(1,2,3,4,5,....,49) 49 in total < 1000
multiples of 5 and 7: 35(1,2,3,4,5,....,28) 28 in total < 1000

199 - 49 - 28 = 122

The double counts include multiples of 20,35 (2*2*5, 7*5)
70,140,350,700,770,910,980

122 + 7(doubles) =129

I think the double counts given by you are incorrect. 70, for example, is not a multiple of 4 and 5.

The double counts are: 140, 280, 420, 560, 700, 840, 980
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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1
Did I understand the concept correctly?

To find multiples of a number X less that Y, we divide Y/X. If Y/X has no remainder then no. of multiples if Y/X - 1. If Y/X has remainder then no. of multiples is quotient of Y/X. Eg

no. of multiples of 5 less than 1000 = 1000/5 - 1=199. (If 1000 was included then we wouldnt have subtracted 1)
No. of multiples of 3 less than 1000 = 1000/3 = 333

To find common multiples of two nos U and V up to some no. W (eg common multiples of 20 and 35 less than 1000), we find the first common multiple (in this case 140). No all multiples of 140 less than 1000 would be common multiples of 20 and 35 less than 1000 (140, 280, 420, 560, 700, 840, 980, total 7)
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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multiple of 5 between 1 to 100 = (100-5)/5 + 1 = 20

multiple of 5&4 that is 20 between 1 to 100 = (100-20)/20 + 1 = 5

multiple of 5&7 that is 35 between 1 to 100 = (100-35)/35 + 1 = 2

Total multiple of 20 and 35 between 1 to 100 = 7

multiple of 5 between 1 to 1000 = 20*10 = 200

multiple of 20 and 35 between 1 to 1000 = 7*10 = 70

multiple of 5 BUT not of 4&7 between 1 to 1000 = 200-70 = 130

we need to subtract 1 because question asks less than 1000.

hence total multiple asked = 130-1 = 129
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GRE 1: Q169 V154 How many positive integers less than 1000 are multiples of 5  [#permalink]

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here is my approach ->
Multiples of 5=> 199
Multiples of 4 and 5 i.e of 20 => 49
Multiples of 7 and 5 i.e 35 => 28
Multiples of all 5,4,& i.e multiples of 140=> 7

Required value => 199-49-28+7 => 129
Smash that D

Unfortunately its taking me over 4 minutes

Any Quickies here abhimahna
Try your hand on this one
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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I am not able to understand why 7 has been added in the end...
Can somebody explain this to me...
Of possible please explain with the help of Venn diagram

Posted from my mobile device
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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[quote="suramya26"]I am not able to understand why 7 has been added in the end...
Can somebody explain this to me...
Of possible please explain with the help of Venn diagram

Hope this will help.

Thanks.
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

Numbers less than 1000 = { 1 , 2 , 3 , 4 ............999 }

Numbers upto 999 Divisible by 5 is $$\frac{999}{5}$$ = 199

Numbers upto 999 Divisible by 20 (Multiple of 5 & 4) is $$\frac{999}{5}$$ = 49
Numbers upto 999 Divisible by 35 (Multiple of 5 & 7) is $$\frac{999}{5}$$ = 28
Numbers upto 999 Divisible by 140 (Multiple of 5 ,4 & 7) is $$\frac{999}{5}$$ = 7
Quote:
So, Total number of positive integers less than 1000 are multiples of 5 but NOT of 4 or 7

=> 199 - 49-28+7 = 129

Hence answer will be (D) 129...

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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

995/5=all multiples of 5=199
980/20=all multiples of 5 and 4=49
980/35=all multiples of 5 and 7=28
980/140=duplicate multiples of 5, 4 and 7=7
199-49-28+7=129
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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1)Multiples of 5: (995-5)/5+1=198+1=199
2)Multiples of 5&4: 20: (980-20)/20+1=960/20+1=48+1=49
3)Multiples of 5&7: 35: (980-35)/35+1=945/35+1=27+1=28
4)Multiples of 5&4&7: 140: (980-140)/140+1=840/140+1=6+1=7
5)Multiples of 5 but not 4 and not 7: 199-(49-7)-(28-7)-7=129
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

Let’s first determine the number of multiples of 5 from 5 to 995, inclusive:

(995 - 5)/5 + 1 = 199

If a number is a multiple of 5, but not a multiple of 4, then it’s not a multiple of 20. So let’s determine the number of multiples of 20 less than 1000:

(980 - 20)/20 + 1 = 49

If a number is a multiple of 5, but not a multiple of 7, then it’s not a multiple of 35. So let’s determine the number of multiples of 35 less than 1000:

(980 - 35)/35 + 1 = 28

Thus, we need to subtract 49 and 28 from 199:

199 - 49 - 28 = 122

However, we “over” subtracted from 199; for example, we subtracted 980 twice, as it is a multiple of 20 and also a multiple of 35. In other words, we “over” subtracted the number of numbers that are multiples of 5, 4 and 7, i.e., a multiple of the LCM of 5, 4, 7, which is 140. So we have to determine the number of multiples of 140 less than 100 and add that number back to 122:

(980 - 140)/140 + 1 = 7

Thus, the number of positive integers less than 1000 which are a multiple of 5 but not 4 or 7 is 122 + 7 = 129.

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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

Was close enough on this one, had to take values less than 1000

No of Multiples of 5 = 199
No of Multiples of 20 = 49
No of Multiples of 35 = 28

Numbers will become 199-49-28 = 122

But i forgot to take the multiples of 4,5 and 7, 140 => 140 * 7= 980

Adding this back to the number will give us the total values = 122 + 7 = 129
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How many positive integers less than 1000 are multiples of 5  [#permalink]

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Number of multiples of 5 ----- 999/5 = 199.xx
Number of multiples of 5&4 ----- 999/20 = 49.xx
Number of multiples of 5&7 ----- 1000/35 = 28.xx
number of multiples of 5&4&7 ( these are repeated twice ) --- 1000/140 = 7.xx

199- 49-28 + 7
129
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

multiples of 5 <1000 ; 199
multiples of 4 & 5 ; i.e 20 <1000 ; 49
multiples of 7& 5 i.e 35 < 1000 ; 28
and multiples of overlap 7,4&5 ; 20&35 ; 140 <1000 ; 7
total integers ; 199+7-49-28 ; 129
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Re: How many positive integers less than 1000 are multiples of 5  [#permalink]

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ScottTargetTestPrep wrote:
kevincan wrote:
How many positive integers less than 1000 are multiples of 5 but NOT of 4 or 7?

(A) 114
(B) 121
(C) 122
(D) 129
(E) 136

Let’s first determine the number of multiples of 5 from 5 to 995, inclusive:

(995 - 5)/5 + 1 = 199

If a number is a multiple of 5, but not a multiple of 4, then it’s not a multiple of 20. So let’s determine the number of multiples of 20 less than 1000:

(980 - 20)/20 + 1 = 49

If a number is a multiple of 5, but not a multiple of 7, then it’s not a multiple of 35. So let’s determine the number of multiples of 35 less than 1000:

(980 - 35)/35 + 1 = 28

Thus, we need to subtract 49 and 28 from 199:

199 - 49 - 28 = 122

However, we “over” subtracted from 199; for example, we subtracted 980 twice, as it is a multiple of 20 and also a multiple of 35. In other words, we “over” subtracted the number of numbers that are multiples of 5, 4 and 7, i.e., a multiple of the LCM of 5, 4, 7, which is 140. So we have to determine the number of multiples of 140 less than 100 and add that number back to 122:

(980 - 140)/140 + 1 = 7

Thus, the number of positive integers less than 1000 which are a multiple of 5 but not 4 or 7 is 122 + 7 = 129.

this line : for example, we subtracted 980 twice, as it is a multiple of 20 and also a multiple of 35.
awesome.....i just wasnt getting why numbers were being subtracted twice..now i do!!

thanks! Re: How many positive integers less than 1000 are multiples of 5   [#permalink] 06 Jan 2020, 01:15
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