To determine the number of integers between 1 and 1000 that are divisible by 2, 5, or 7, we can use the Principle of Inclusion-Exclusion. Let A be the set of integers divisible by 2, B be the set divisible by 5, and C be the set divisible by 7.
- |A| = Number of integers divisible by 2 = 1000/2 = 500
- |B| = Number of integers divisible by 5 = 1000/5 = 200
- |C| = Number of integers divisible by 7 = 1000/7 = 142 (integer part)
- |A ∩ B| = Number of integers divisible by 2 and 5 (i.e., by 10) = 1000/10 = 100
- |A ∩ C| = Number of integers divisible by 2 and 7 (i.e., by 14) = 1000/14 = 71 (integer part)
- |B ∩ C| = Number of integers divisible by 5 and 7 (i.e., by 35) = 1000/35 = 28 (integer part)
- |A ∩ B ∩ C| = Number of integers divisible by 2, 5, and 7 (i.e., by 70) = 1000/70 = 14 (integer part)
Using the Principle of Inclusion-Exclusion:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
|A ∪ B ∪ C| = 500 + 200 + 142 - 100 - 71 - 28 + 14
|A ∪ B ∪ C| = 657
Therefore, there are 657 integers between 1 and 1000 that are divisible by 2, 5, or 7.
It's important to note that the provided reference excerpt discusses numbers divisible by 2, 3, 5, or 7 and mentions specific numbers 210, 420, 630, and 840. However, the main question asked here is about divisibility by 2, 5, or 7, so the excerpt is not directly relevant to the calculation.
