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How many positive integers less than 1000 are multiples of 5
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Updated on: 30 Apr 2015, 03:34
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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
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Originally posted by kevincan on 14 Apr 2007, 23:34.
Last edited by Bunuel on 30 Apr 2015, 03:34, edited 1 time in total.
Renamed the topic, edited the question and added the OA.




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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 06:48
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, 1994928=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.




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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 01:42
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 20078 = 122 multiples.
So the answere is C.
Javed.
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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 06:49
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?



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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 10:08
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, 1994928=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.
I agree with your explanation. Answere is 129.
Javed
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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 12:12
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



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Re: How many positive integers less than 1000 are multiples of 5
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15 Apr 2007, 13:51
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
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15 Jun 2016, 23:50
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
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18 Jun 2016, 07:15
multiple of 5 between 1 to 100 = (1005)/5 + 1 = 20 multiple of 5&4 that is 20 between 1 to 100 = (10020)/20 + 1 = 5 multiple of 5&7 that is 35 between 1 to 100 = (10035)/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 = 20070 = 130 we need to subtract 1 because question asks less than 1000.hence total multiple asked = 1301 = 129
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How many positive integers less than 1000 are multiples of 5
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23 Aug 2016, 10:58
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 => 1994928+7 => 129 Smash that D Unfortunately its taking me over 4 minutes Any Quickies here abhimahnaTry your hand on this one
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Re: How many positive integers less than 1000 are multiples of 5
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13 Oct 2016, 03:07
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
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Re: How many positive integers less than 1000 are multiples of 5
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13 Oct 2016, 04:01
[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
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13 Oct 2016, 08:28
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  4928+7 = 129 Hence answer will be (D) 129...
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Re: How many positive integers less than 1000 are multiples of 5
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14 Oct 2016, 15:04
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 1994928+7=129 D.



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Re: How many positive integers less than 1000 are multiples of 5
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20 Jan 2017, 18:37
1)Multiples of 5: (9955)/5+1=198+1=199 2)Multiples of 5&4: 20: (98020)/20+1=960/20+1=48+1=49 3)Multiples of 5&7: 35: (98035)/35+1=945/35+1=27+1=28 4)Multiples of 5&4&7: 140: (980140)/140+1=840/140+1=6+1=7 5)Multiples of 5 but not 4 and not 7: 199(497)(287)7=129



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Re: How many positive integers less than 1000 are multiples of 5
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05 Feb 2018, 10:20
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. Answer: D
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Re: How many positive integers less than 1000 are multiples of 5
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03 Feb 2019, 08:02
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 1994928 = 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
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03 Feb 2019, 21:55
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 4928 + 7 129 Answer is D



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Re: How many positive integers less than 1000 are multiples of 5
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30 Jul 2019, 07:43
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+74928 ; 129 IMO D




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