The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ________.

Let $U = \{1, 2, \dots, 500\}$, so $|U| = 500$.

Let:

• $A$ be the set of integers divisible by 3

• $B$ be the set of integers divisible by 5

• $C$ be the set of integers divisible by 7

Using the Principle of Inclusion-Exclusion (PIE):

Calculating individual cardinalities:

• $|A| = \lfloor \frac{500}{3} \rfloor = 166$

• $|B| = \lfloor \frac{500}{5} \rfloor = 100$

• $|C| = \lfloor \frac{500}{7} \rfloor = 71$

Calculating pairwise intersections:

• $|A \cap B| = \lfloor \frac{500}{\text{lcm}(3, 5)} \rfloor = \lfloor \frac{500}{15} \rfloor = 33$

• $|B \cap C| = \lfloor \frac{500}{\text{lcm}(5, 7)} \rfloor = \lfloor \frac{500}{35} \rfloor = 14$

• $|C \cap A| = \lfloor \frac{500}{\text{lcm}(7, 3)} \rfloor = \lfloor \frac{500}{21} \rfloor = 23$

Calculating three-way intersection:

• $|A \cap B \cap C| = \lfloor \frac{500}{\text{lcm}(3, 5, 7)} \rfloor = \lfloor \frac{500}{105} \rfloor = 4$

Applying PIE:

Thus, the answer is 271 [cite: 2].