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Inclusion-Exclusion (Easy)

AMC 10 Q1–8 depth — contest entry, not school drills.

Easy Medium Hard AIME / USAJMO / AMO

Inclusion-Exclusion: core idea

Two sets.

Example problem: hint and answer

How many integers from $1$ to $30$ are divisible by $2$ **or** $3$ ?

Hint: $\lfloor30/2\rfloor+\lfloor30/3\rfloor-\lfloor30/6\rfloor$ . Answer: 20.

Practice (9 problems — type your answers)

1. How many integers from $1$ to $50$ are divisible by $2$ **or** $5$ ?
2. Sets $A,B$ satisfy $|A|=20$ , $|B|=15$ , $|A\cap B|=8$ . Find $|A\cup B|$ .
3. How many integers from $1$ to $60$ are divisible by $3$ **or** $4$ ?
4. How many integers from $1$ to $24$ are divisible by $2$ **or** $3$ ?
5. Sets $A,B$ satisfy $|A|=30$ , $|B|=22$ , $|A\cap B|=10$ . Find $|A\cup B|$ .
6. How many integers from $1$ to $45$ are divisible by $3$ **or** $5$ ?
7. How many integers from $1$ to $100$ are divisible by $2$ **or** $5$ ?
8. How many integers from $1$ to $36$ are divisible by $2$ **or** $3$ ?
9. Sets $A,B$ satisfy $|A|=12$ , $|B|=18$ , $|A\cap B|=5$ . Find $|A\cup B|$ .

Inclusion-Exclusion Practice Problems — Easy