What are the divisors of 1040?

1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 40, 52, 65, 80, 104, 130, 208, 260, 520, 1040

There is a total of 20 positive divisors.
The sum of these divisors is 2604.
The arithmetic mean is 130.2.

16 even divisors

2, 4, 8, 10, 16, 20, 26, 40, 52, 80, 104, 130, 208, 260, 520, 1040

4 odd divisors

1, 5, 13, 65

How to compute the divisors of 1040?

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 1040 by each of the numbers from 1 to 1040 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)

1040 / 1 = 1040 (the remainder is 0, so 1 is a divisor of 1040)
1040 / 2 = 520 (the remainder is 0, so 2 is a divisor of 1040)
1040 / 3 = 346.66666666667 (the remainder is 2, so 3 is not a divisor of 1040)
...
1040 / 1039 = 1.0009624639076 (the remainder is 1, so 1039 is not a divisor of 1040)
1040 / 1040 = 1 (the remainder is 0, so 1040 is a divisor of 1040)

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 1040 (i.e. 32.249030993194). 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 $\frac{N}{d} = D$ )

1040 / 1 = 1040 (the remainder is 0, so 1 and 1040 are divisors of 1040)
1040 / 2 = 520 (the remainder is 0, so 2 and 520 are divisors of 1040)
1040 / 3 = 346.66666666667 (the remainder is 2, so 3 is not a divisor of 1040)
...
1040 / 31 = 33.548387096774 (the remainder is 17, so 31 is not a divisor of 1040)
1040 / 32 = 32.5 (the remainder is 16, so 32 is not a divisor of 1040)