What are the divisors of 5616?

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 36, 39, 48, 52, 54, 72, 78, 104, 108, 117, 144, 156, 208, 216, 234, 312, 351, 432, 468, 624, 702, 936, 1404, 1872, 2808, 5616

There is a total of 40 positive divisors.
The sum of these divisors is 17360.
The arithmetic mean is 434.

32 even divisors

2, 4, 6, 8, 12, 16, 18, 24, 26, 36, 48, 52, 54, 72, 78, 104, 108, 144, 156, 208, 216, 234, 312, 432, 468, 624, 702, 936, 1404, 1872, 2808, 5616

8 odd divisors

1, 3, 9, 13, 27, 39, 117, 351

How to compute the divisors of 5616?

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 5616 by each of the numbers from 1 to 5616 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)

5616 / 1 = 5616 (the remainder is 0, so 1 is a divisor of 5616)
5616 / 2 = 2808 (the remainder is 0, so 2 is a divisor of 5616)
5616 / 3 = 1872 (the remainder is 0, so 3 is a divisor of 5616)
...
5616 / 5615 = 1.00017809439 (the remainder is 1, so 5615 is not a divisor of 5616)
5616 / 5616 = 1 (the remainder is 0, so 5616 is a divisor of 5616)

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 5616 (i.e. 74.939975980781). 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 $\frac{N}{d} = D$ )

5616 / 1 = 5616 (the remainder is 0, so 1 and 5616 are divisors of 5616)
5616 / 2 = 2808 (the remainder is 0, so 2 and 2808 are divisors of 5616)
5616 / 3 = 1872 (the remainder is 0, so 3 and 1872 are divisors of 5616)
...
5616 / 73 = 76.931506849315 (the remainder is 68, so 73 is not a divisor of 5616)
5616 / 74 = 75.891891891892 (the remainder is 66, so 74 is not a divisor of 5616)