How many positive integers less than 1000 are multiples of 5

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

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)

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

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



Unfortunately its taking me over 4 minutes

Any Quickies here abhimahna
Try your hand on this one

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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[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.

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


=> 199 - 49-28+7 = 129

Hence answer will be (D) 129...

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
D.

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

First lets count the multiple of 5 between 1 to 100 which is 20 (5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100)

Out of these multiples there are 5 numbers which are multiples of 5 and 4 and similarily there are 2 no which are mutiples of both 5 & 7. Hence we need to subtract total 7 from 20 which comes to 13.

Now till 1000 numbers from 1 to 100 will come 10 times so we can multiple 13 by 10 which comes to 130 . Out of these 130 number one will be extra which is 1000 so we can subtract 130 -1 = 129 Ans D

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