What are the divisors of 756?
1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
- There is a total of 24 positive divisors.
- The sum of these divisors is 2240.
- The arithmetic mean is 93.333333333333.
16 even divisors
2, 4, 6, 12, 14, 18, 28, 36, 42, 54, 84, 108, 126, 252, 378, 756
8 odd divisors
1, 3, 7, 9, 21, 27, 63, 189
How to compute the divisors of 756?
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 756 by each of the numbers from 1 to 756 to determine which ones have a remainder equal to 0.
(where is the integer part of the quotient)
- 756 / 1 = 756 (the remainder is 0, so 1 is a divisor of 756)
- 756 / 2 = 378 (the remainder is 0, so 2 is a divisor of 756)
- 756 / 3 = 252 (the remainder is 0, so 3 is a divisor of 756)
- ...
- 756 / 755 = 1.0013245033113 (the remainder is 1, so 755 is not a divisor of 756)
- 756 / 756 = 1 (the remainder is 0, so 756 is a divisor of 756)
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 756 (i.e. 27.495454169735). 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 )
- 756 / 1 = 756 (the remainder is 0, so 1 and 756 are divisors of 756)
- 756 / 2 = 378 (the remainder is 0, so 2 and 378 are divisors of 756)
- 756 / 3 = 252 (the remainder is 0, so 3 and 252 are divisors of 756)
- ...
- 756 / 26 = 29.076923076923 (the remainder is 2, so 26 is not a divisor of 756)
- 756 / 27 = 28 (the remainder is 0, so 27 and 28 are divisors of 756)