What are the divisors of 264?
1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264
- There is a total of 16 positive divisors.
- The sum of these divisors is 720.
- The arithmetic mean is 45.
12 even divisors
2, 4, 6, 8, 12, 22, 24, 44, 66, 88, 132, 264
4 odd divisors
1, 3, 11, 33
How to compute the divisors of 264?
A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.
Brute force algorithm
We could start by using a brute-force method which would involve dividing 264 by each of the numbers from 1 to 264 to determine which ones have a remainder equal to 0.
(where is the integer part of the quotient)
- 264 / 1 = 264 (the remainder is 0, so 1 is a divisor of 264)
- 264 / 2 = 132 (the remainder is 0, so 2 is a divisor of 264)
- 264 / 3 = 88 (the remainder is 0, so 3 is a divisor of 264)
- ...
- 264 / 263 = 1.0038022813688 (the remainder is 1, so 263 is not a divisor of 264)
- 264 / 264 = 1 (the remainder is 0, so 264 is a divisor of 264)
Improved algorithm using square-root
However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 264 (i.e. 16.248076809272). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:
(thus, if , then )
- 264 / 1 = 264 (the remainder is 0, so 1 and 264 are divisors of 264)
- 264 / 2 = 132 (the remainder is 0, so 2 and 132 are divisors of 264)
- 264 / 3 = 88 (the remainder is 0, so 3 and 88 are divisors of 264)
- ...
- 264 / 15 = 17.6 (the remainder is 9, so 15 is not a divisor of 264)
- 264 / 16 = 16.5 (the remainder is 8, so 16 is not a divisor of 264)