What are the divisors of 684?

1, 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684

There is a total of 18 positive divisors.
The sum of these divisors is 1820.
The arithmetic mean is 101.11111111111.

12 even divisors

2, 4, 6, 12, 18, 36, 38, 76, 114, 228, 342, 684

6 odd divisors

1, 3, 9, 19, 57, 171

How to compute the divisors of 684?

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 684 by each of the numbers from 1 to 684 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)

684 / 1 = 684 (the remainder is 0, so 1 is a divisor of 684)
684 / 2 = 342 (the remainder is 0, so 2 is a divisor of 684)
684 / 3 = 228 (the remainder is 0, so 3 is a divisor of 684)
...
684 / 683 = 1.0014641288433 (the remainder is 1, so 683 is not a divisor of 684)
684 / 684 = 1 (the remainder is 0, so 684 is a divisor of 684)

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 684 (i.e. 26.153393661244). 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 $\frac{N}{d} = D$ )

684 / 1 = 684 (the remainder is 0, so 1 and 684 are divisors of 684)
684 / 2 = 342 (the remainder is 0, so 2 and 342 are divisors of 684)
684 / 3 = 228 (the remainder is 0, so 3 and 228 are divisors of 684)
...
684 / 25 = 27.36 (the remainder is 9, so 25 is not a divisor of 684)
684 / 26 = 26.307692307692 (the remainder is 8, so 26 is not a divisor of 684)