What are the divisors of 8191?

1, 8191

There is a total of 2 positive divisors.
The sum of these divisors is 8192.
The arithmetic mean is 4096.

2 odd divisors

1, 8191

How to compute the divisors of 8191?

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 8191 by each of the numbers from 1 to 8191 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)

8191 / 1 = 8191 (the remainder is 0, so 1 is a divisor of 8191)
8191 / 2 = 4095.5 (the remainder is 1, so 2 is not a divisor of 8191)
8191 / 3 = 2730.3333333333 (the remainder is 1, so 3 is not a divisor of 8191)
...
8191 / 8190 = 1.0001221001221 (the remainder is 1, so 8190 is not a divisor of 8191)
8191 / 8191 = 1 (the remainder is 0, so 8191 is a divisor of 8191)

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 8191 (i.e. 90.504143551552). 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 $\frac{N}{d} = D$ )

8191 / 1 = 8191 (the remainder is 0, so 1 and 8191 are divisors of 8191)
8191 / 2 = 4095.5 (the remainder is 1, so 2 is not a divisor of 8191)
8191 / 3 = 2730.3333333333 (the remainder is 1, so 3 is not a divisor of 8191)
...
8191 / 89 = 92.033707865169 (the remainder is 3, so 89 is not a divisor of 8191)
8191 / 90 = 91.011111111111 (the remainder is 1, so 90 is not a divisor of 8191)