Exercise 1000Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Elements with none of the properties

#IE#complement#none

Statement

Among $100$ integers, $50$ are even, $33$ divisible by $3$ , $20$ by $5$ , $16$ by $6$ , $10$ by $10$ , $6$ by $15$ , $3$ by $30$ . How many are coprime to $30$ ?

Hint

Compute union first (multiples of $2$ , $3$ , $5$ ), then complement.

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Solution

Union. $50+33+20-16-10-6+3=74$ .

Complement. $100-74=26$ .

Conclusion. $26$ integers coprime to $30$ .

Elements with none of the properties

Statement

Solution

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