Question

find number of integers between 1 and 10000 inclusive which are divisible by none of 5 or6 Or 8
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Answer 1

Step-by-step explanation:

आरम्भ करने के लिए दिल्ली परिवहन डीटीसी के महाप्रभु

are divisible by none of 5 or6 Or 8number of integers between 1 and 10000 inclusive which are divisible by none of 5 or6 Or 8 okkk for points

Answer 2

Given,

Series of numbers between 1 and 10,000.

To find,

The numbers in the given series that are not divisible by 5 or 6 or 8.

Solution,

Let the Sample set be,

$S=${$1,2,3,...,10000$}

And let,

$A_{1}=$ set of numbers divisible by 5

$A_{2}=$ set of numbers divisible by 6

$A_{3}=$ set of numbers divisible by 8

Now, we need to find the numbers which are not included in $A_{1},A_{2},A_{3}$

So,

$|A_{1}|= \frac{10000}{5}$

⇒$|A_{1}|=2500$

Similarly,

$|A_{2}|=\frac{10000}{6}$

⇒$|A_{2}|=1666$

And,

$|A_{3}|=\frac{10000}{8}$

⇒$|A_{3}|=1250$

Now,

To find the numbers in S that are multiples of both 5 and 6, we need to find lcm of them.

So,

$LCM(5,6)=30$

$LCM(6,8)=24$

$LCM(5,8)=40$

$LCM(5,6,8)=120$

Now,

$|A_{1}$∩$A_{2}|=\frac{10000}{30}$

⇒$|A_{1}$∩$A_{2}|=333$

And,

$|A_{2}$∩$A_{3}|=\frac{10000}{24}$

⇒$|A_{2}$∩$A_{3}|=416$

Also,

$|A_{1}$∩$A_{3}|=\frac{10000}{40}$

⇒$|A_{1}$∩$A_{3}|=250$

And Hence,

$|A_{1}$∩$A_{2}$∩$A_{3}|=\frac{10000}{120}$

⇒$|A_{1}$∩$A_{2}$∩$A_{3}|=83$

Therefore,

Total numbers in $S$ that are not divisible by 5 or 6 or 8 are,

⇒$X=10000-(2500+1666+1250)+(333+416+250)-83$

⇒$X=10000-5416+999-83$

⇒$X=10999-5499$

⇒$X=5500$

Hence, total numbers between 1 and 10,000 that are not divisible by 5 or 6 or 8 are 5,500.