What are the divisors of 687?

1, 3, 229, 687

There is a total of 4 positive divisors.
The sum of these divisors is 920.
The arithmetic mean is 230.

4 odd divisors

1, 3, 229, 687

How to compute the divisors of 687?

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 687 by each of the numbers from 1 to 687 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)

687 / 1 = 687 (the remainder is 0, so 1 is a divisor of 687)
687 / 2 = 343.5 (the remainder is 1, so 2 is not a divisor of 687)
687 / 3 = 229 (the remainder is 0, so 3 is a divisor of 687)
...
687 / 686 = 1.0014577259475 (the remainder is 1, so 686 is not a divisor of 687)
687 / 687 = 1 (the remainder is 0, so 687 is a divisor of 687)

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 687 (i.e. 26.210684844162). 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 $\frac{N}{d} = D$ )

687 / 1 = 687 (the remainder is 0, so 1 and 687 are divisors of 687)
687 / 2 = 343.5 (the remainder is 1, so 2 is not a divisor of 687)
687 / 3 = 229 (the remainder is 0, so 3 and 229 are divisors of 687)
...
687 / 25 = 27.48 (the remainder is 12, so 25 is not a divisor of 687)
687 / 26 = 26.423076923077 (the remainder is 11, so 26 is not a divisor of 687)