What are the divisors of 1038?

1, 2, 3, 6, 173, 346, 519, 1038

There is a total of 8 positive divisors.
The sum of these divisors is 2088.
The arithmetic mean is 261.

4 even divisors

2, 6, 346, 1038

4 odd divisors

1, 3, 173, 519

How to compute the divisors of 1038?

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 1038 by each of the numbers from 1 to 1038 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)

1038 / 1 = 1038 (the remainder is 0, so 1 is a divisor of 1038)
1038 / 2 = 519 (the remainder is 0, so 2 is a divisor of 1038)
1038 / 3 = 346 (the remainder is 0, so 3 is a divisor of 1038)
...
1038 / 1037 = 1.0009643201543 (the remainder is 1, so 1037 is not a divisor of 1038)
1038 / 1038 = 1 (the remainder is 0, so 1038 is a divisor of 1038)

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 1038 (i.e. 32.218007387174). 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 $\frac{N}{d} = D$ )

1038 / 1 = 1038 (the remainder is 0, so 1 and 1038 are divisors of 1038)
1038 / 2 = 519 (the remainder is 0, so 2 and 519 are divisors of 1038)
1038 / 3 = 346 (the remainder is 0, so 3 and 346 are divisors of 1038)
...
1038 / 31 = 33.483870967742 (the remainder is 15, so 31 is not a divisor of 1038)
1038 / 32 = 32.4375 (the remainder is 14, so 32 is not a divisor of 1038)