What are the divisors of 1039?

1, 1039

There is a total of 2 positive divisors.
The sum of these divisors is 1040.
The arithmetic mean is 520.

2 odd divisors

1, 1039

How to compute the divisors of 1039?

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 1039 by each of the numbers from 1 to 1039 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)

1039 / 1 = 1039 (the remainder is 0, so 1 is a divisor of 1039)
1039 / 2 = 519.5 (the remainder is 1, so 2 is not a divisor of 1039)
1039 / 3 = 346.33333333333 (the remainder is 1, so 3 is not a divisor of 1039)
...
1039 / 1038 = 1.0009633911368 (the remainder is 1, so 1038 is not a divisor of 1039)
1039 / 1039 = 1 (the remainder is 0, so 1039 is a divisor of 1039)

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 1039 (i.e. 32.233522922572). 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 $\frac{N}{d} = D$ )

1039 / 1 = 1039 (the remainder is 0, so 1 and 1039 are divisors of 1039)
1039 / 2 = 519.5 (the remainder is 1, so 2 is not a divisor of 1039)
1039 / 3 = 346.33333333333 (the remainder is 1, so 3 is not a divisor of 1039)
...
1039 / 31 = 33.516129032258 (the remainder is 16, so 31 is not a divisor of 1039)
1039 / 32 = 32.46875 (the remainder is 15, so 32 is not a divisor of 1039)