What are the divisors of 5760?

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 32, 36, 40, 45, 48, 60, 64, 72, 80, 90, 96, 120, 128, 144, 160, 180, 192, 240, 288, 320, 360, 384, 480, 576, 640, 720, 960, 1152, 1440, 1920, 2880, 5760

There is a total of 48 positive divisors.
The sum of these divisors is 19890.
The arithmetic mean is 414.375.

42 even divisors

2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 90, 96, 120, 128, 144, 160, 180, 192, 240, 288, 320, 360, 384, 480, 576, 640, 720, 960, 1152, 1440, 1920, 2880, 5760

6 odd divisors

1, 3, 5, 9, 15, 45

How to compute the divisors of 5760?

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 5760 by each of the numbers from 1 to 5760 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)

5760 / 1 = 5760 (the remainder is 0, so 1 is a divisor of 5760)
5760 / 2 = 2880 (the remainder is 0, so 2 is a divisor of 5760)
5760 / 3 = 1920 (the remainder is 0, so 3 is a divisor of 5760)
...
5760 / 5759 = 1.0001736412572 (the remainder is 1, so 5759 is not a divisor of 5760)
5760 / 5760 = 1 (the remainder is 0, so 5760 is a divisor of 5760)

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 5760 (i.e. 75.894663844041). 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 $\frac{N}{d} = D$ )

5760 / 1 = 5760 (the remainder is 0, so 1 and 5760 are divisors of 5760)
5760 / 2 = 2880 (the remainder is 0, so 2 and 2880 are divisors of 5760)
5760 / 3 = 1920 (the remainder is 0, so 3 and 1920 are divisors of 5760)
...
5760 / 74 = 77.837837837838 (the remainder is 62, so 74 is not a divisor of 5760)
5760 / 75 = 76.8 (the remainder is 60, so 75 is not a divisor of 5760)