What are the divisors of 685?

1, 5, 137, 685

There is a total of 4 positive divisors.
The sum of these divisors is 828.
The arithmetic mean is 207.

4 odd divisors

1, 5, 137, 685

How to compute the divisors of 685?

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 685 by each of the numbers from 1 to 685 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)

685 / 1 = 685 (the remainder is 0, so 1 is a divisor of 685)
685 / 2 = 342.5 (the remainder is 1, so 2 is not a divisor of 685)
685 / 3 = 228.33333333333 (the remainder is 1, so 3 is not a divisor of 685)
...
685 / 684 = 1.0014619883041 (the remainder is 1, so 684 is not a divisor of 685)
685 / 685 = 1 (the remainder is 0, so 685 is a divisor of 685)

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 685 (i.e. 26.172504656605). 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 $\frac{N}{d} = D$ )

685 / 1 = 685 (the remainder is 0, so 1 and 685 are divisors of 685)
685 / 2 = 342.5 (the remainder is 1, so 2 is not a divisor of 685)
685 / 3 = 228.33333333333 (the remainder is 1, so 3 is not a divisor of 685)
...
685 / 25 = 27.4 (the remainder is 10, so 25 is not a divisor of 685)
685 / 26 = 26.346153846154 (the remainder is 9, so 26 is not a divisor of 685)