What are the divisors of 713?

1, 23, 31, 713

There is a total of 4 positive divisors.
The sum of these divisors is 768.
The arithmetic mean is 192.

4 odd divisors

1, 23, 31, 713

How to compute the divisors of 713?

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 713 by each of the numbers from 1 to 713 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)

713 / 1 = 713 (the remainder is 0, so 1 is a divisor of 713)
713 / 2 = 356.5 (the remainder is 1, so 2 is not a divisor of 713)
713 / 3 = 237.66666666667 (the remainder is 2, so 3 is not a divisor of 713)
...
713 / 712 = 1.001404494382 (the remainder is 1, so 712 is not a divisor of 713)
713 / 713 = 1 (the remainder is 0, so 713 is a divisor of 713)

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 713 (i.e. 26.702059845637). 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 $\frac{N}{d} = D$ )

713 / 1 = 713 (the remainder is 0, so 1 and 713 are divisors of 713)
713 / 2 = 356.5 (the remainder is 1, so 2 is not a divisor of 713)
713 / 3 = 237.66666666667 (the remainder is 2, so 3 is not a divisor of 713)
...
713 / 25 = 28.52 (the remainder is 13, so 25 is not a divisor of 713)
713 / 26 = 27.423076923077 (the remainder is 11, so 26 is not a divisor of 713)