What are the divisors of 3564?

1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 27, 33, 36, 44, 54, 66, 81, 99, 108, 132, 162, 198, 297, 324, 396, 594, 891, 1188, 1782, 3564

There is a total of 30 positive divisors.
The sum of these divisors is 10164.
The arithmetic mean is 338.8.

20 even divisors

2, 4, 6, 12, 18, 22, 36, 44, 54, 66, 108, 132, 162, 198, 324, 396, 594, 1188, 1782, 3564

10 odd divisors

1, 3, 9, 11, 27, 33, 81, 99, 297, 891

How to compute the divisors of 3564?

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 3564 by each of the numbers from 1 to 3564 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)

3564 / 1 = 3564 (the remainder is 0, so 1 is a divisor of 3564)
3564 / 2 = 1782 (the remainder is 0, so 2 is a divisor of 3564)
3564 / 3 = 1188 (the remainder is 0, so 3 is a divisor of 3564)
...
3564 / 3563 = 1.0002806623632 (the remainder is 1, so 3563 is not a divisor of 3564)
3564 / 3564 = 1 (the remainder is 0, so 3564 is a divisor of 3564)

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 3564 (i.e. 59.699246226397). 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 $\frac{N}{d} = D$ )

3564 / 1 = 3564 (the remainder is 0, so 1 and 3564 are divisors of 3564)
3564 / 2 = 1782 (the remainder is 0, so 2 and 1782 are divisors of 3564)
3564 / 3 = 1188 (the remainder is 0, so 3 and 1188 are divisors of 3564)
...
3564 / 58 = 61.448275862069 (the remainder is 26, so 58 is not a divisor of 3564)
3564 / 59 = 60.406779661017 (the remainder is 24, so 59 is not a divisor of 3564)