What are the divisors of 8144?

1, 2, 4, 8, 16, 509, 1018, 2036, 4072, 8144

There is a total of 10 positive divisors.
The sum of these divisors is 15810.
The arithmetic mean is 1581.

8 even divisors

2, 4, 8, 16, 1018, 2036, 4072, 8144

2 odd divisors

1, 509

How to compute the divisors of 8144?

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 8144 by each of the numbers from 1 to 8144 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)

8144 / 1 = 8144 (the remainder is 0, so 1 is a divisor of 8144)
8144 / 2 = 4072 (the remainder is 0, so 2 is a divisor of 8144)
8144 / 3 = 2714.6666666667 (the remainder is 2, so 3 is not a divisor of 8144)
...
8144 / 8143 = 1.0001228048631 (the remainder is 1, so 8143 is not a divisor of 8144)
8144 / 8144 = 1 (the remainder is 0, so 8144 is a divisor of 8144)

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 8144 (i.e. 90.244113381428). 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 $\frac{N}{d} = D$ )

8144 / 1 = 8144 (the remainder is 0, so 1 and 8144 are divisors of 8144)
8144 / 2 = 4072 (the remainder is 0, so 2 and 4072 are divisors of 8144)
8144 / 3 = 2714.6666666667 (the remainder is 2, so 3 is not a divisor of 8144)
...
8144 / 89 = 91.505617977528 (the remainder is 45, so 89 is not a divisor of 8144)
8144 / 90 = 90.488888888889 (the remainder is 44, so 90 is not a divisor of 8144)