What are the divisors of 1022?

1, 2, 7, 14, 73, 146, 511, 1022

There is a total of 8 positive divisors.
The sum of these divisors is 1776.
The arithmetic mean is 222.

4 even divisors

2, 14, 146, 1022

4 odd divisors

1, 7, 73, 511

How to compute the divisors of 1022?

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 1022 by each of the numbers from 1 to 1022 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)

1022 / 1 = 1022 (the remainder is 0, so 1 is a divisor of 1022)
1022 / 2 = 511 (the remainder is 0, so 2 is a divisor of 1022)
1022 / 3 = 340.66666666667 (the remainder is 2, so 3 is not a divisor of 1022)
...
1022 / 1021 = 1.0009794319295 (the remainder is 1, so 1021 is not a divisor of 1022)
1022 / 1022 = 1 (the remainder is 0, so 1022 is a divisor of 1022)

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 1022 (i.e. 31.968734726292). 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 $\frac{N}{d} = D$ )

1022 / 1 = 1022 (the remainder is 0, so 1 and 1022 are divisors of 1022)
1022 / 2 = 511 (the remainder is 0, so 2 and 511 are divisors of 1022)
1022 / 3 = 340.66666666667 (the remainder is 2, so 3 is not a divisor of 1022)
...
1022 / 30 = 34.066666666667 (the remainder is 2, so 30 is not a divisor of 1022)
1022 / 31 = 32.967741935484 (the remainder is 30, so 31 is not a divisor of 1022)