What are the divisors of 721?

1, 7, 103, 721

There is a total of 4 positive divisors.
The sum of these divisors is 832.
The arithmetic mean is 208.

4 odd divisors

1, 7, 103, 721

How to compute the divisors of 721?

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 721 by each of the numbers from 1 to 721 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)

721 / 1 = 721 (the remainder is 0, so 1 is a divisor of 721)
721 / 2 = 360.5 (the remainder is 1, so 2 is not a divisor of 721)
721 / 3 = 240.33333333333 (the remainder is 1, so 3 is not a divisor of 721)
...
721 / 720 = 1.0013888888889 (the remainder is 1, so 720 is not a divisor of 721)
721 / 721 = 1 (the remainder is 0, so 721 is a divisor of 721)

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 721 (i.e. 26.851443164195). 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 $\frac{N}{d} = D$ )

721 / 1 = 721 (the remainder is 0, so 1 and 721 are divisors of 721)
721 / 2 = 360.5 (the remainder is 1, so 2 is not a divisor of 721)
721 / 3 = 240.33333333333 (the remainder is 1, so 3 is not a divisor of 721)
...
721 / 25 = 28.84 (the remainder is 21, so 25 is not a divisor of 721)
721 / 26 = 27.730769230769 (the remainder is 19, so 26 is not a divisor of 721)