discrete math functions pigeonhole principle How many numbers in

Dylan Yoder 2022-04-14 Answered

discrete math 
functions pigeonhole principle
How many numbers in the set {1,2,3,,346} are divisible by 5 or 7 ?

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Answers (1)

szalbierzfytg
Answered 2022-04-15 Author has 13 answers

In the given problem we have to find how many numbers in the set {1,2,3,,346} are divisible by 5 or 7.
Denote A= {1,2,3,,346}
Suppose A5 be the subset of A containing those numbers which are divisible by 5 and
A7 be the subset of A containing those numbers which are divisible by 7.
Then A5A7 contain those numbers of A which are divisible by both 5 and 7. 
And a number is divisible by both 5 and 7 if it is divisible by their least common multiple and lcm(5,7)=35 .
Therefore a number is divisible by both 5 and 7 if it is divisible by 35.
A5A7 contain those numbers of A which are divisible by both 5 or 7.
Therefore we have
n(A5)=[3465]=69 where [] is the greatest integer function.
and
n(A7)=[3467]=49
and
n(A5A7)=[34635]=9
Now
n(A5A7)=n(A5)+(A7)n(A5A7)
n(A5A7)=69+499
n(A5A7)=109
Hence there are 109 numbers in the set {1,2,3,,346} are divisible by 5 or 7.

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