There are exactly 450 odd positive integers between 100 and 999 inclusive.
Understanding the Range
To accurately determine the count of odd numbers, it's crucial to first identify the total number of integers within the specified range. The question asks for positive integers between 100 and 999 inclusive, which means both 100 and 999 are part of the set we are considering.
- Start of the range: 100
- End of the range: 999
Calculating the Total Number of Integers
To find the total count of integers in an inclusive range [A, B], the formula is B - A + 1.
- Subtract the start from the end:
999 - 100 = 899 - Add 1 for inclusivity:
899 + 1 = 900
Therefore, there are 900 positive integers between 100 and 999 inclusive.
Determining the Number of Odd Integers
A fundamental principle in number theory is that in any continuous sequence of integers, if the total count of numbers is even, then exactly half of them will be odd, and half will be even.
As stated by Vaia, "Half of the numbers in any range are odd. Since there are 900 numbers between 100 and 999, half of them are odd."
- Total numbers in the range: 900
- Applying the principle: Since 900 is an even number, we can simply divide the total by 2 to find the number of odd integers.
Calculation:
900 / 2 = 450
Thus, there are 450 odd numbers within this range. This holds true because the range starts with an even number (100) and ends with an odd number (999), encompassing an equal distribution of odd and even numbers.
To summarize the calculation:
| Metric | Value |
|---|---|
| Starting Integer | 100 |
| Ending Integer | 999 |
| Total Integers (999-100+1) | 900 |
| Odd Integers (900 / 2) | 450 |
