What are the divisors of 8192?

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

There is a total of 14 positive divisors.
The sum of these divisors is 16383.
The arithmetic mean is 1170.2142857143.

13 even divisors

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

1 odd divisors

How to compute the divisors of 8192?

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 8192 by each of the numbers from 1 to 8192 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)

8192 / 1 = 8192 (the remainder is 0, so 1 is a divisor of 8192)
8192 / 2 = 4096 (the remainder is 0, so 2 is a divisor of 8192)
8192 / 3 = 2730.6666666667 (the remainder is 2, so 3 is not a divisor of 8192)
...
8192 / 8191 = 1.0001220852155 (the remainder is 1, so 8191 is not a divisor of 8192)
8192 / 8192 = 1 (the remainder is 0, so 8192 is a divisor of 8192)

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 8192 (i.e. 90.509667991878). 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 $\frac{N}{d} = D$ )

8192 / 1 = 8192 (the remainder is 0, so 1 and 8192 are divisors of 8192)
8192 / 2 = 4096 (the remainder is 0, so 2 and 4096 are divisors of 8192)
8192 / 3 = 2730.6666666667 (the remainder is 2, so 3 is not a divisor of 8192)
...
8192 / 89 = 92.044943820225 (the remainder is 4, so 89 is not a divisor of 8192)
8192 / 90 = 91.022222222222 (the remainder is 2, so 90 is not a divisor of 8192)