What are the divisors of 724?

1, 2, 4, 181, 362, 724

There is a total of 6 positive divisors.
The sum of these divisors is 1274.
The arithmetic mean is 212.33333333333.

4 even divisors

2, 4, 362, 724

2 odd divisors

1, 181

How to compute the divisors of 724?

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 724 by each of the numbers from 1 to 724 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)

724 / 1 = 724 (the remainder is 0, so 1 is a divisor of 724)
724 / 2 = 362 (the remainder is 0, so 2 is a divisor of 724)
724 / 3 = 241.33333333333 (the remainder is 1, so 3 is not a divisor of 724)
...
724 / 723 = 1.0013831258645 (the remainder is 1, so 723 is not a divisor of 724)
724 / 724 = 1 (the remainder is 0, so 724 is a divisor of 724)

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 724 (i.e. 26.907248094147). 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 $\frac{N}{d} = D$ )

724 / 1 = 724 (the remainder is 0, so 1 and 724 are divisors of 724)
724 / 2 = 362 (the remainder is 0, so 2 and 362 are divisors of 724)
724 / 3 = 241.33333333333 (the remainder is 1, so 3 is not a divisor of 724)
...
724 / 25 = 28.96 (the remainder is 24, so 25 is not a divisor of 724)
724 / 26 = 27.846153846154 (the remainder is 22, so 26 is not a divisor of 724)