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Let and denote the sets of positive integers that are divisible by 2, 3 and 5 respectively and less or equal to 100.
If and are relatively prime then a is divisible by and iff it is divisible by
It follows that and
Therefore, by Inclusion-exclusion principle, the number of positive integers that are divisible by 2 or 3 or 5 and less or equal to 100 is equal to
We conclude that the number of integers that are not divisible by 2 or 3 or 5 is equal to