There are 50 odd numbers between 1 and 100, inclusive.
Understanding Odd Numbers
Odd numbers are integers that cannot be divided exactly by 2. When an odd number is divided by 2, it always leaves a remainder of 1. These numbers always end in 1, 3, 5, 7, or 9. Conversely, even numbers are integers perfectly divisible by 2, ending in 0, 2, 4, 6, or 8.
Numbers naturally alternate between odd and even in sequence. For example, starting from 1: 1 (odd), 2 (even), 3 (odd), 4 (even), and so on. This consistent pattern helps in easily determining the count of odd or even numbers within a given range. To learn more about the properties of odd and even numbers, you can refer to Math is Fun: Odd and Even Numbers.
Counting Odd Numbers from 1 to 100
Given the range from 1 to 100, both inclusive, we are looking at a consecutive sequence of 100 numbers. Since the sequence starts with an odd number (1) and ends with an even number (100), the numbers are perfectly distributed between odd and even.
Here's how to confirm the count:
- Identify the first and last odd numbers: The first odd number in the range is 1, and the last odd number is 99.
- Recognize the pattern: Odd numbers form an arithmetic progression where each term increases by 2.
- Apply the formula (optional but illustrative):
For an arithmetic sequence, the number of terms ($n$) can be found using the formula:
$n = (\text{Last Term} - \text{First Term}) / \text{Common Difference} + 1$
In this case:
$n = (99 - 1) / 2 + 1$
$n = 98 / 2 + 1$
$n = 49 + 1$
$n = 50$
Therefore, there are exactly 50 odd numbers within this range. This means that for every two numbers in the sequence, one is odd and one is even, leading to an equal split for a range that starts with an odd number and ends with an even number like 1 to 100.
Illustrative List of Odd Numbers (Partial)
| Odd Numbers |
|---|
| 1 |
| 3 |
| 5 |
| ... |
| 95 |
| 97 |
| 99 |
Identifying Odd Numbers Quickly
To quickly determine if a number is odd, simply look at its last digit.
- A number is odd if its last digit is:
- 1
- 3
- 5
- 7
- 9
For example, 73 is odd because it ends in 3. The number 96 is even because it ends in 6. This simple rule applies to numbers of any size.
