There are 666 integer numbers from 1 to 1000 inclusive that are divisible by 2 or by 3.
To find this answer, we use the principle of inclusion-exclusion. This principle helps to avoid double-counting numbers when dealing with sets that have overlapping members. Here's the breakdown:
Counting Multiples
- Multiples of 2: There are 1000 / 2 = 500 numbers divisible by 2 between 1 and 1000.
- Multiples of 3: There are 1000 / 3 = 333.33, so 333 numbers divisible by 3 between 1 and 1000 (we only count whole numbers).
Avoiding Double Counting
Numbers divisible by both 2 and 3 are multiples of their least common multiple, which is 6.
- Multiples of 6: 1000 / 6 = 166.66, so there are 166 numbers divisible by 6. As per the reference provided, there are 166 integers between 1 & 1000 that are divisible by both 2 & 3.
Inclusion-Exclusion Principle
To find the number of integers divisible by 2 or 3, we add the number of multiples of 2 and the number of multiples of 3, and then subtract the number of multiples of 6 (which we counted twice):
Total = (Multiples of 2) + (Multiples of 3) - (Multiples of 6)
Total = 500 + 333 - 166
Total = 667
The Solution
Therefore, the total number of integer numbers from 1 to 1000 that are divisible by either 2 or 3 is 667.
Summary Table
| Category | Count |
|---|---|
| Multiples of 2 | 500 |
| Multiples of 3 | 333 |
| Multiples of 6 | 166 |
| Total (2 or 3) | 667 |
