What are the divisors of 1024?

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

There is a total of 11 positive divisors.
The sum of these divisors is 2047.
The arithmetic mean is 186.09090909091.

10 even divisors

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

1 odd divisors

How to compute the divisors of 1024?

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 1024 by each of the numbers from 1 to 1024 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)

1024 / 1 = 1024 (the remainder is 0, so 1 is a divisor of 1024)
1024 / 2 = 512 (the remainder is 0, so 2 is a divisor of 1024)
1024 / 3 = 341.33333333333 (the remainder is 1, so 3 is not a divisor of 1024)
...
1024 / 1023 = 1.0009775171065 (the remainder is 1, so 1023 is not a divisor of 1024)
1024 / 1024 = 1 (the remainder is 0, so 1024 is a divisor of 1024)

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 1024 (i.e. 32). 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 $\frac{N}{d} = D$ )

1024 / 1 = 1024 (the remainder is 0, so 1 and 1024 are divisors of 1024)
1024 / 2 = 512 (the remainder is 0, so 2 and 512 are divisors of 1024)
1024 / 3 = 341.33333333333 (the remainder is 1, so 3 is not a divisor of 1024)
...
1024 / 31 = 33.032258064516 (the remainder is 1, so 31 is not a divisor of 1024)
1024 / 32 = 32 (the remainder is 0, so 32 and 32 are divisors of 1024)