What are the divisors of 2184?

1, 2, 3, 4, 6, 7, 8, 12, 13, 14, 21, 24, 26, 28, 39, 42, 52, 56, 78, 84, 91, 104, 156, 168, 182, 273, 312, 364, 546, 728, 1092, 2184

There is a total of 32 positive divisors.
The sum of these divisors is 6720.
The arithmetic mean is 210.

24 even divisors

2, 4, 6, 8, 12, 14, 24, 26, 28, 42, 52, 56, 78, 84, 104, 156, 168, 182, 312, 364, 546, 728, 1092, 2184

8 odd divisors

1, 3, 7, 13, 21, 39, 91, 273

How to compute the divisors of 2184?

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 2184 by each of the numbers from 1 to 2184 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)

2184 / 1 = 2184 (the remainder is 0, so 1 is a divisor of 2184)
2184 / 2 = 1092 (the remainder is 0, so 2 is a divisor of 2184)
2184 / 3 = 728 (the remainder is 0, so 3 is a divisor of 2184)
...
2184 / 2183 = 1.0004580852038 (the remainder is 1, so 2183 is not a divisor of 2184)
2184 / 2184 = 1 (the remainder is 0, so 2184 is a divisor of 2184)

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 2184 (i.e. 46.733285782192). 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 $\frac{N}{d} = D$ )

2184 / 1 = 2184 (the remainder is 0, so 1 and 2184 are divisors of 2184)
2184 / 2 = 1092 (the remainder is 0, so 2 and 1092 are divisors of 2184)
2184 / 3 = 728 (the remainder is 0, so 3 and 728 are divisors of 2184)
...
2184 / 45 = 48.533333333333 (the remainder is 24, so 45 is not a divisor of 2184)
2184 / 46 = 47.478260869565 (the remainder is 22, so 46 is not a divisor of 2184)