What are the divisors of 728?

1, 2, 4, 7, 8, 13, 14, 26, 28, 52, 56, 91, 104, 182, 364, 728

There is a total of 16 positive divisors.
The sum of these divisors is 1680.
The arithmetic mean is 105.

12 even divisors

2, 4, 8, 14, 26, 28, 52, 56, 104, 182, 364, 728

4 odd divisors

1, 7, 13, 91

How to compute the divisors of 728?

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.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 728 by each of the numbers from 1 to 728 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

728 / 1 = 728 (the remainder is 0, so 1 is a divisor of 728)
728 / 2 = 364 (the remainder is 0, so 2 is a divisor of 728)
728 / 3 = 242.66666666667 (the remainder is 2, so 3 is not a divisor of 728)
...
728 / 727 = 1.0013755158184 (the remainder is 1, so 727 is not a divisor of 728)
728 / 728 = 1 (the remainder is 0, so 728 is a divisor of 728)

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 728 (i.e. 26.981475126464). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

728 / 1 = 728 (the remainder is 0, so 1 and 728 are divisors of 728)
728 / 2 = 364 (the remainder is 0, so 2 and 364 are divisors of 728)
728 / 3 = 242.66666666667 (the remainder is 2, so 3 is not a divisor of 728)
...
728 / 25 = 29.12 (the remainder is 3, so 25 is not a divisor of 728)
728 / 26 = 28 (the remainder is 0, so 26 and 28 are divisors of 728)