To solve the problem of finding the number of integers between 1 and 1000 divisible by 2, 3, or 5, we use the inclusion-exclusion principle:

Step 1: Calculate individual sets

• (A): Divisible by 2 → (\lfloor \frac{1000}{2} \rfloor = 500)
• (B): Divisible by 3 → (\lfloor \frac{1000}{3} \rfloor = 333)
• (C): Divisible by 5 → (\lfloor \frac{1000}{5} \rfloor = 200)

Step 2: Calculate pairwise intersections

• (A∩B): Divisible by 6 → (\lfloor \frac{1000}{6} \rfloor = 166)
• (A∩C): Divisible by 10 → (\lfloor \frac{1000}{10} \rfloor = 100)
• (B∩C): Divisible by 15 → (\lfloor \frac{1000}{15} \rfloor = 66)

Step 3: Calculate triple intersection

• (A∩B∩C): Divisible by 30 → (\lfloor \frac{1000}{30} \rfloor = 33)

Step 4: Apply inclusion-exclusion

[ |A∪B∪C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C| ]
[ = 500 + 333 + 200 - 166 - 100 - 66 + 33 = 734 ]

Answer: (\boxed{734})

作者声明:本文包含人工智能生成内容。