What are the divisors of 1059?

1, 3, 353, 1059

There is a total of 4 positive divisors.
The sum of these divisors is 1416.
The arithmetic mean is 354.

4 odd divisors

1, 3, 353, 1059

How to compute the divisors of 1059?

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 1059 by each of the numbers from 1 to 1059 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)

1059 / 1 = 1059 (the remainder is 0, so 1 is a divisor of 1059)
1059 / 2 = 529.5 (the remainder is 1, so 2 is not a divisor of 1059)
1059 / 3 = 353 (the remainder is 0, so 3 is a divisor of 1059)
...
1059 / 1058 = 1.0009451795841 (the remainder is 1, so 1058 is not a divisor of 1059)
1059 / 1059 = 1 (the remainder is 0, so 1059 is a divisor of 1059)

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 1059 (i.e. 32.542280190546). 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 $\frac{N}{d} = D$ )

1059 / 1 = 1059 (the remainder is 0, so 1 and 1059 are divisors of 1059)
1059 / 2 = 529.5 (the remainder is 1, so 2 is not a divisor of 1059)
1059 / 3 = 353 (the remainder is 0, so 3 and 353 are divisors of 1059)
...
1059 / 31 = 34.161290322581 (the remainder is 5, so 31 is not a divisor of 1059)
1059 / 32 = 33.09375 (the remainder is 3, so 32 is not a divisor of 1059)