What are the divisors of 1023?

1, 3, 11, 31, 33, 93, 341, 1023

There is a total of 8 positive divisors.
The sum of these divisors is 1536.
The arithmetic mean is 192.

8 odd divisors

1, 3, 11, 31, 33, 93, 341, 1023

How to compute the divisors of 1023?

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 1023 by each of the numbers from 1 to 1023 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)

1023 / 1 = 1023 (the remainder is 0, so 1 is a divisor of 1023)
1023 / 2 = 511.5 (the remainder is 1, so 2 is not a divisor of 1023)
1023 / 3 = 341 (the remainder is 0, so 3 is a divisor of 1023)
...
1023 / 1022 = 1.0009784735812 (the remainder is 1, so 1022 is not a divisor of 1023)
1023 / 1023 = 1 (the remainder is 0, so 1023 is a divisor of 1023)

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 1023 (i.e. 31.984371183439). 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 $\frac{N}{d} = D$ )

1023 / 1 = 1023 (the remainder is 0, so 1 and 1023 are divisors of 1023)
1023 / 2 = 511.5 (the remainder is 1, so 2 is not a divisor of 1023)
1023 / 3 = 341 (the remainder is 0, so 3 and 341 are divisors of 1023)
...
1023 / 30 = 34.1 (the remainder is 3, so 30 is not a divisor of 1023)
1023 / 31 = 33 (the remainder is 0, so 31 and 33 are divisors of 1023)