What are the divisors of 723?

1, 3, 241, 723

There is a total of 4 positive divisors.
The sum of these divisors is 968.
The arithmetic mean is 242.

4 odd divisors

1, 3, 241, 723

How to compute the divisors of 723?

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 723 by each of the numbers from 1 to 723 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)

723 / 1 = 723 (the remainder is 0, so 1 is a divisor of 723)
723 / 2 = 361.5 (the remainder is 1, so 2 is not a divisor of 723)
723 / 3 = 241 (the remainder is 0, so 3 is a divisor of 723)
...
723 / 722 = 1.0013850415512 (the remainder is 1, so 722 is not a divisor of 723)
723 / 723 = 1 (the remainder is 0, so 723 is a divisor of 723)

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 723 (i.e. 26.888659319498). 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 $\frac{N}{d} = D$ )

723 / 1 = 723 (the remainder is 0, so 1 and 723 are divisors of 723)
723 / 2 = 361.5 (the remainder is 1, so 2 is not a divisor of 723)
723 / 3 = 241 (the remainder is 0, so 3 and 241 are divisors of 723)
...
723 / 25 = 28.92 (the remainder is 23, so 25 is not a divisor of 723)
723 / 26 = 27.807692307692 (the remainder is 21, so 26 is not a divisor of 723)