Find the number of positive integers not exceeding 100 that

Find the number of positive integers not exceeding 100 that are not divisible by 5 or by 7.

Answer :

Let us take the set A = {1, 2, …. , 100} ,

B={nA ; 5 divides n}= {n A ; 5 |n} then |B| = 20

and C = {n A ; 7 divides n}={n A ; 7| n} then |C| = 14

So that we must find |A – (B C)|

since B and C are not disjoint, so |B C |= |B| +|C| |B C|.

As to B C, its elements are those numbers in A such that are divisible by 35 (that is, only 35 and 70).

Therefore, |B C| = 2. Hence, |B C |= |B| +|C| |B C|= 20 + 14 2 = 32

Therefore |A – (B C)| = |A| |(B C)| = 100 – 32 = 68.

So, there are exactly 68 numbers not exceeding 100 that are not divisible by 5 or by 7.