Exercise 988Combinatorics › Inclusion-exclusionL1Added 3 Nov 2026

Coprime to $100$

#IE#coprime#totient

Statement

How many integers from $1$ to $100$ are coprime to $100$ ?

Hint

$100=2^2\cdot5^2$ , so coprime means not divisible by $2$ or $5$ .

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Sets. $A_2$ (multiples of $2$ ): $50$ ; $A_5$ (multiples of $5$ ): $20$ ; $A_2\cap A_5$ (multiples of $10$ ): $10$ .

Union. $50+20-10=60$ .

Complement. $100-60=40$ .

Conclusion. $40$ coprime to $100$ (matching $\varphi(100)=40$ ).