What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 n...........

Mathematics (NDA) Set theory

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What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

860

854

840

824

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Explanation:

Let A be the set of natural numbers ( $\leq$ 1000)which are divisible by 10,
B be the set of natural numbers ( $\leq$ 1000) which are divisible by 15 and
C be the set of natural numbers ( $\leq$ 1000) which are divisible by 25.
Then, n(A) = [1000/10]=100; n(B)= [1000/15]=66 and
n(C) =[1000/25]=40, where [. ] denotes the greatest
integer function.
Now, n(A $\cap$ B) = n (set of numbers which are divisible by both 10 and 15)
= n (set of numbers divisible by 30) = [1000/30]=33
Similarly, n(A $\cap$ C) = n(set of numbers divisible by 50)
= [1000/50]=20
n(B C) = n (set of numbers divisible by 75)= [1000/75]=13
and n(A $\cap$ B $\cap$ C) = n (set of numbers divisible by 10,
15 and 25)
= n (set of numbers divisible by 150)=[1000/150]=6
$\therefore$ n(A $\cup$ B $\cup$ C) = n(A) + n(B) + n(C) − n(A $\cap$ B)
− n(B $\cap$ C) − n(C $\cap$ A) + n(A $\cap$ B $\cap$ C)
= 100 + 66 + 40 − 33 − 13 − 20 + 6 = 146
Hence, required numbers = n(U) − n(A $\cup$ B $\cup$ C)
= 1000 − 146 = 854