What are the divisors of 5520?

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 23, 24, 30, 40, 46, 48, 60, 69, 80, 92, 115, 120, 138, 184, 230, 240, 276, 345, 368, 460, 552, 690, 920, 1104, 1380, 1840, 2760, 5520

There is a total of 40 positive divisors.
The sum of these divisors is 17856.
The arithmetic mean is 446.4.

32 even divisors

2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 40, 46, 48, 60, 80, 92, 120, 138, 184, 230, 240, 276, 368, 460, 552, 690, 920, 1104, 1380, 1840, 2760, 5520

8 odd divisors

1, 3, 5, 15, 23, 69, 115, 345

How to compute the divisors of 5520?

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 5520 by each of the numbers from 1 to 5520 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)

5520 / 1 = 5520 (the remainder is 0, so 1 is a divisor of 5520)
5520 / 2 = 2760 (the remainder is 0, so 2 is a divisor of 5520)
5520 / 3 = 1840 (the remainder is 0, so 3 is a divisor of 5520)
...
5520 / 5519 = 1.000181192245 (the remainder is 1, so 5519 is not a divisor of 5520)
5520 / 5520 = 1 (the remainder is 0, so 5520 is a divisor of 5520)

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 5520 (i.e. 74.296702484027). 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 $\frac{N}{d} = D$ )

5520 / 1 = 5520 (the remainder is 0, so 1 and 5520 are divisors of 5520)
5520 / 2 = 2760 (the remainder is 0, so 2 and 2760 are divisors of 5520)
5520 / 3 = 1840 (the remainder is 0, so 3 and 1840 are divisors of 5520)
...
5520 / 73 = 75.616438356164 (the remainder is 45, so 73 is not a divisor of 5520)
5520 / 74 = 74.594594594595 (the remainder is 44, so 74 is not a divisor of 5520)