What are the divisors of 8224?

1, 2, 4, 8, 16, 32, 257, 514, 1028, 2056, 4112, 8224

There is a total of 12 positive divisors.
The sum of these divisors is 16254.
The arithmetic mean is 1354.5.

10 even divisors

2, 4, 8, 16, 32, 514, 1028, 2056, 4112, 8224

2 odd divisors

1, 257

How to compute the divisors of 8224?

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 8224 by each of the numbers from 1 to 8224 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)

8224 / 1 = 8224 (the remainder is 0, so 1 is a divisor of 8224)
8224 / 2 = 4112 (the remainder is 0, so 2 is a divisor of 8224)
8224 / 3 = 2741.3333333333 (the remainder is 1, so 3 is not a divisor of 8224)
...
8224 / 8223 = 1.000121610118 (the remainder is 1, so 8223 is not a divisor of 8224)
8224 / 8224 = 1 (the remainder is 0, so 8224 is a divisor of 8224)

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 8224 (i.e. 90.686272390037). 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 $\frac{N}{d} = D$ )

8224 / 1 = 8224 (the remainder is 0, so 1 and 8224 are divisors of 8224)
8224 / 2 = 4112 (the remainder is 0, so 2 and 4112 are divisors of 8224)
8224 / 3 = 2741.3333333333 (the remainder is 1, so 3 is not a divisor of 8224)
...
8224 / 89 = 92.404494382022 (the remainder is 36, so 89 is not a divisor of 8224)
8224 / 90 = 91.377777777778 (the remainder is 34, so 90 is not a divisor of 8224)