What are the divisors of 683?

1, 683

There is a total of 2 positive divisors.
The sum of these divisors is 684.
The arithmetic mean is 342.

2 odd divisors

1, 683

How to compute the divisors of 683?

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 683 by each of the numbers from 1 to 683 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)

683 / 1 = 683 (the remainder is 0, so 1 is a divisor of 683)
683 / 2 = 341.5 (the remainder is 1, so 2 is not a divisor of 683)
683 / 3 = 227.66666666667 (the remainder is 2, so 3 is not a divisor of 683)
...
683 / 682 = 1.0014662756598 (the remainder is 1, so 682 is not a divisor of 683)
683 / 683 = 1 (the remainder is 0, so 683 is a divisor of 683)

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 683 (i.e. 26.134268690744). 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 $\frac{N}{d} = D$ )

683 / 1 = 683 (the remainder is 0, so 1 and 683 are divisors of 683)
683 / 2 = 341.5 (the remainder is 1, so 2 is not a divisor of 683)
683 / 3 = 227.66666666667 (the remainder is 2, so 3 is not a divisor of 683)
...
683 / 25 = 27.32 (the remainder is 8, so 25 is not a divisor of 683)
683 / 26 = 26.269230769231 (the remainder is 7, so 26 is not a divisor of 683)