What are the divisors of 1030?

1, 2, 5, 10, 103, 206, 515, 1030

There is a total of 8 positive divisors.
The sum of these divisors is 1872.
The arithmetic mean is 234.

4 even divisors

2, 10, 206, 1030

4 odd divisors

1, 5, 103, 515

How to compute the divisors of 1030?

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 1030 by each of the numbers from 1 to 1030 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)

1030 / 1 = 1030 (the remainder is 0, so 1 is a divisor of 1030)
1030 / 2 = 515 (the remainder is 0, so 2 is a divisor of 1030)
1030 / 3 = 343.33333333333 (the remainder is 1, so 3 is not a divisor of 1030)
...
1030 / 1029 = 1.0009718172983 (the remainder is 1, so 1029 is not a divisor of 1030)
1030 / 1030 = 1 (the remainder is 0, so 1030 is a divisor of 1030)

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 1030 (i.e. 32.093613071762). 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 $\frac{N}{d} = D$ )

1030 / 1 = 1030 (the remainder is 0, so 1 and 1030 are divisors of 1030)
1030 / 2 = 515 (the remainder is 0, so 2 and 515 are divisors of 1030)
1030 / 3 = 343.33333333333 (the remainder is 1, so 3 is not a divisor of 1030)
...
1030 / 31 = 33.225806451613 (the remainder is 7, so 31 is not a divisor of 1030)
1030 / 32 = 32.1875 (the remainder is 6, so 32 is not a divisor of 1030)