What are the divisors of 1028?

1, 2, 4, 257, 514, 1028

There is a total of 6 positive divisors.
The sum of these divisors is 1806.
The arithmetic mean is 301.

4 even divisors

2, 4, 514, 1028

2 odd divisors

1, 257

How to compute the divisors of 1028?

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 1028 by each of the numbers from 1 to 1028 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)

1028 / 1 = 1028 (the remainder is 0, so 1 is a divisor of 1028)
1028 / 2 = 514 (the remainder is 0, so 2 is a divisor of 1028)
1028 / 3 = 342.66666666667 (the remainder is 2, so 3 is not a divisor of 1028)
...
1028 / 1027 = 1.0009737098345 (the remainder is 1, so 1027 is not a divisor of 1028)
1028 / 1028 = 1 (the remainder is 0, so 1028 is a divisor of 1028)

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 1028 (i.e. 32.062439083763). 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 $\frac{N}{d} = D$ )

1028 / 1 = 1028 (the remainder is 0, so 1 and 1028 are divisors of 1028)
1028 / 2 = 514 (the remainder is 0, so 2 and 514 are divisors of 1028)
1028 / 3 = 342.66666666667 (the remainder is 2, so 3 is not a divisor of 1028)
...
1028 / 31 = 33.161290322581 (the remainder is 5, so 31 is not a divisor of 1028)
1028 / 32 = 32.125 (the remainder is 4, so 32 is not a divisor of 1028)