What are the divisors of 1045?

1, 5, 11, 19, 55, 95, 209, 1045

There is a total of 8 positive divisors.
The sum of these divisors is 1440.
The arithmetic mean is 180.

8 odd divisors

1, 5, 11, 19, 55, 95, 209, 1045

How to compute the divisors of 1045?

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 1045 by each of the numbers from 1 to 1045 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)

1045 / 1 = 1045 (the remainder is 0, so 1 is a divisor of 1045)
1045 / 2 = 522.5 (the remainder is 1, so 2 is not a divisor of 1045)
1045 / 3 = 348.33333333333 (the remainder is 1, so 3 is not a divisor of 1045)
...
1045 / 1044 = 1.0009578544061 (the remainder is 1, so 1044 is not a divisor of 1045)
1045 / 1045 = 1 (the remainder is 0, so 1045 is a divisor of 1045)

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 1045 (i.e. 32.326459750489). 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 $\frac{N}{d} = D$ )

1045 / 1 = 1045 (the remainder is 0, so 1 and 1045 are divisors of 1045)
1045 / 2 = 522.5 (the remainder is 1, so 2 is not a divisor of 1045)
1045 / 3 = 348.33333333333 (the remainder is 1, so 3 is not a divisor of 1045)
...
1045 / 31 = 33.709677419355 (the remainder is 22, so 31 is not a divisor of 1045)
1045 / 32 = 32.65625 (the remainder is 21, so 32 is not a divisor of 1045)