What are the divisors of 1440?

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 32, 36, 40, 45, 48, 60, 72, 80, 90, 96, 120, 144, 160, 180, 240, 288, 360, 480, 720, 1440

There is a total of 36 positive divisors.
The sum of these divisors is 4914.
The arithmetic mean is 136.5.

30 even divisors

2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 72, 80, 90, 96, 120, 144, 160, 180, 240, 288, 360, 480, 720, 1440

6 odd divisors

1, 3, 5, 9, 15, 45

How to compute the divisors of 1440?

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 1440 by each of the numbers from 1 to 1440 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)

1440 / 1 = 1440 (the remainder is 0, so 1 is a divisor of 1440)
1440 / 2 = 720 (the remainder is 0, so 2 is a divisor of 1440)
1440 / 3 = 480 (the remainder is 0, so 3 is a divisor of 1440)
...
1440 / 1439 = 1.0006949270327 (the remainder is 1, so 1439 is not a divisor of 1440)
1440 / 1440 = 1 (the remainder is 0, so 1440 is a divisor of 1440)

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 1440 (i.e. 37.947331922021). 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 $\frac{N}{d} = D$ )

1440 / 1 = 1440 (the remainder is 0, so 1 and 1440 are divisors of 1440)
1440 / 2 = 720 (the remainder is 0, so 2 and 720 are divisors of 1440)
1440 / 3 = 480 (the remainder is 0, so 3 and 480 are divisors of 1440)
...
1440 / 36 = 40 (the remainder is 0, so 36 and 40 are divisors of 1440)
1440 / 37 = 38.918918918919 (the remainder is 34, so 37 is not a divisor of 1440)