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Exercise 883Combinatorics › Counting principlesL1Added 3 Nov 2026

Counting with at least one

#inclusion-exclusion#at least one#complement

Statement

How many numbers from 11 to 100100 are divisible by 22, 33, or 55?

Hint

Use inclusion-exclusion on the three sets.

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Solution

Sets.
- ∣A2∣=⌊100/2⌋=50|A_2|=\lfloor100/2\rfloor=50
- ∣A3∣=⌊100/3⌋=33|A_3|=\lfloor100/3\rfloor=33
- ∣A5∣=⌊100/5⌋=20|A_5|=\lfloor100/5\rfloor=20
- ∣A2∩A3∣=∣A6∣=16|A_2\cap A_3|=|A_6|=16
- ∣A2∩A5∣=∣A10∣=10|A_2\cap A_5|=|A_{10}|=10
- ∣A3∩A5∣=∣A15∣=6|A_3\cap A_5|=|A_{15}|=6
- ∣A2∩A3∩A5∣=∣A30∣=3|A_2\cap A_3\cap A_5|=|A_{30}|=3

Union. 50+33+20−16−10−6+3=7450+33+20-16-10-6+3=74.

Conclusion. 7474 numbers.