Question

Find the number of integers between 1 & 1000, inclusive that are not divisible by 5,6,8​

Answer 1

Answer:

Step-by-step explanation: number of integers between 1 and 1000 divisible by 10 is 100

number of integers between 1 and 1000 divisible by 15 is 66

number of integers between 1 and 1000 divisible by 25 is 40

number of integers between 1 and 1000 divisible by 10 and 15 is multiples of 30 and is 33

number of integers between 1 and 1000 divisible by 10 and 25 is multiples of 50 and is 20

number of integers between 1 and 1000 divisible by 25 and 15 is multiples of 75 and is 13

number of integers between 1 and 1000 divisible by 10,15 and 25 is multiples of 150 and is 6

number of integers between 1 and 1000 (both inclusive) which are divisible by either of 10, 15 or 25 is

100+66+40=206

now 206-(33+20+13)+6

=146 Ans

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Answer 2

Answer:

The number of integers between 1 & 1000, inclusive that are not divisible by 5,6,8​ are 600

Step-by-step explanation:

Let the number of integers divisible by 5 be represented as n(5)

Let the number of integers divisible by 6 be represented as n(6)

Let the number of integers divisible by 8 be represented as n(8)

Let the number of integers divisible by both 5 and 6 be represented as n(5∩6)

Let the number of integers divisible by both 6 and 8 be represented as n(6∩8)

Let the number of integers divisible by both 5 and 8 be represented as n(5∩8)

Let the number of integers divisible 5,6 and 8 be represented as n(5∩6∩8)

First, lets find all the numbers that are divisible by 5,6,8 which is represented as n(5∪6∪8)

General Formula:

n(A∪B∪C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)

n(5∪6∪8)= n(5)+n(6)+n(8)-n(5∩6)-n(6∩8)-n(5∩8)+n(5∩6∩8)-------( 1 )

To find n(5):

The numbers divisible by 5 from 1 to 1000 are 5,10,15,20.....1000

So, a=5 and d=5

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$1000=5+(n-1)5\\1000=5+5n-5\\1000=5n\\n=200$

∴The number of integers divisible by 5 from 1 to 1000 are, n(5)= 200

---(2)

To find n(6):

The numbers divisible by 6 from 1 to 1000 are 6,12,18,24,,,,996

So, a=6 and d=6

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$996=6+(n-1)6\\996=6+6n-6\\996=6n\\n=166$

∴The number of integers divisible by 6 from 1 to 1000 are, n(6)= 166

---(3)

To find n(8):

The numbers divisible by 8 from 1 to 1000 are 8,16,24,32...1000

So, a=8 and d=8

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$1000=8+(n-1)8\\1000=8+8n-8\\1000=8n\\n=125$

∴The number of integers divisible by 8 from 1 to 1000 are, n(8)= 125

---(4)

To find n(5∩6):

The numbers divisible by 5 and 6 from 1 to 1000 would be the LCM of 5 and 6 (i.e.)30 and the numbers would be 30,60,90,.....990

So, a=30 and d=30

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$990=30+(n-1)30\\990=30+30n-30\\990=30n\\n=33$

∴The number of integers divisible by 30 from 1 to 1000 are, n(5∩6)= 33---(5)

To find n(6∩8):

The numbers divisible by 6 and 8 from 1 to 1000 would be the LCM of 6 and 8 (i.e.)24 and the numbers would be 24,48,72,96,....984

So, a=24 and d=24

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$984=24+(n-1)24\\984=24+24n-24\\984=24n\\n=41$

∴The number of integers divisible by 24 from 1 to 1000 are, n(6∩8)= 41

----(6)

To find n(5∩8):

The numbers divisible by 5 and 8 from 1 to 1000 would be the LCM of 5 and 8 (i.e.)40 and the numbers would be 40,80,120,....1000

So, a=40 and d=40

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$1000=40+(n-1)40\\1000=40+40n-40\\1000=40n\\n=25$

∴The number of integers divisible by 40 from 1 to 1000 are, n(5∩8)= 25---(7)

To find n(5∩6∩8):

The numbers divisible by 5, 6 and 8 from 1 to 1000 would be the LCM of 5, 6 and 8 (i.e.)120 and the numbers would be 120,240,360,480....,960

So, a=120 and d=120

Finding the $n^{th}$ term:

$n^{th}= a+(n-1)d$

$960=120+(n-1)120\\960=120+120n-120\\960=120n\\n=8$

∴The number of integers divisible by 120 from 1 to 1000 are, n(5∩6∩8)= 8---(8)

Substituting the values in equation 2, 3, 4, 5, 6, 7, 8 in equation (1) we get,

n(5∪6∪8)= n(5)+n(6)+n(8)-n(5∩6)-n(6∩8)-n(5∩8)+n(5∩6∩8)

n(5∪6∪8) = 200+166+125-33-41-25+8

n(5∪6∪8) =400

∴ The number of integers from 1 to 1000 that are divisible by 5, 6 or 8 =400

Required number of integers between 1 to 1000 that are not divisible by 5,6 and 8 are:

                       = 1000 - n(5∪6∪8)

                       =1000- 400

                       =600

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