There are 31 numbers less than 1000 that have an odd number of factors.
Understanding Factors and Perfect Squares
The number of factors a number has is determined by its prime factorization. Factors usually come in pairs. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. We can pair them as (1, 12), (2, 6), and (3, 4).
However, for perfect squares, one factor is paired with itself. For example, the factors of 9 are 1, 3, and 9. The pairs are (1, 9) and (3, 3). Because 3 is paired with itself, the number of factors is odd.
Identifying Perfect Squares Less Than 1000
Therefore, only perfect squares have an odd number of factors. We need to find how many perfect squares are less than 1000.
We need to find the largest integer n such that n2 < 1000.
- 12 = 1
- 22 = 4
- 32 = 9
- ...
- 302 = 900
- 312 = 961
- 322 = 1024
Since 312 = 961 is less than 1000, and 322 = 1024 is greater than 1000, the largest perfect square less than 1000 is 961, which is the square of 31.
Therefore, the perfect squares less than 1000 are 12, 22, 32, ..., 312. There are 31 such numbers.
Conclusion
There are 31 numbers less than 1000 that have an odd number of factors. These are the perfect squares from 1 to 31.
