What are the divisors of 440?

1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440

There is a total of 16 positive divisors.
The sum of these divisors is 1080.
The arithmetic mean is 67.5.

12 even divisors

2, 4, 8, 10, 20, 22, 40, 44, 88, 110, 220, 440

4 odd divisors

1, 5, 11, 55

How to compute the divisors of 440?

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 440 by each of the numbers from 1 to 440 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)

440 / 1 = 440 (the remainder is 0, so 1 is a divisor of 440)
440 / 2 = 220 (the remainder is 0, so 2 is a divisor of 440)
440 / 3 = 146.66666666667 (the remainder is 2, so 3 is not a divisor of 440)
...
440 / 439 = 1.002277904328 (the remainder is 1, so 439 is not a divisor of 440)
440 / 440 = 1 (the remainder is 0, so 440 is a divisor of 440)

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 440 (i.e. 20.976176963403). 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 $\frac{N}{d} = D$ )

440 / 1 = 440 (the remainder is 0, so 1 and 440 are divisors of 440)
440 / 2 = 220 (the remainder is 0, so 2 and 220 are divisors of 440)
440 / 3 = 146.66666666667 (the remainder is 2, so 3 is not a divisor of 440)
...
440 / 19 = 23.157894736842 (the remainder is 3, so 19 is not a divisor of 440)
440 / 20 = 22 (the remainder is 0, so 20 and 22 are divisors of 440)