What are the divisors of 2040?

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 17, 20, 24, 30, 34, 40, 51, 60, 68, 85, 102, 120, 136, 170, 204, 255, 340, 408, 510, 680, 1020, 2040

There is a total of 32 positive divisors.
The sum of these divisors is 6480.
The arithmetic mean is 202.5.

24 even divisors

2, 4, 6, 8, 10, 12, 20, 24, 30, 34, 40, 60, 68, 102, 120, 136, 170, 204, 340, 408, 510, 680, 1020, 2040

8 odd divisors

1, 3, 5, 15, 17, 51, 85, 255

How to compute the divisors of 2040?

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 2040 by each of the numbers from 1 to 2040 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)

2040 / 1 = 2040 (the remainder is 0, so 1 is a divisor of 2040)
2040 / 2 = 1020 (the remainder is 0, so 2 is a divisor of 2040)
2040 / 3 = 680 (the remainder is 0, so 3 is a divisor of 2040)
...
2040 / 2039 = 1.0004904364885 (the remainder is 1, so 2039 is not a divisor of 2040)
2040 / 2040 = 1 (the remainder is 0, so 2040 is a divisor of 2040)

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 2040 (i.e. 45.166359162545). 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 $\frac{N}{d} = D$ )

2040 / 1 = 2040 (the remainder is 0, so 1 and 2040 are divisors of 2040)
2040 / 2 = 1020 (the remainder is 0, so 2 and 1020 are divisors of 2040)
2040 / 3 = 680 (the remainder is 0, so 3 and 680 are divisors of 2040)
...
2040 / 44 = 46.363636363636 (the remainder is 16, so 44 is not a divisor of 2040)
2040 / 45 = 45.333333333333 (the remainder is 15, so 45 is not a divisor of 2040)