What are the divisors of 8093?

1, 8093

There is a total of 2 positive divisors.
The sum of these divisors is 8094.
The arithmetic mean is 4047.

2 odd divisors

1, 8093

How to compute the divisors of 8093?

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 8093 by each of the numbers from 1 to 8093 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)

8093 / 1 = 8093 (the remainder is 0, so 1 is a divisor of 8093)
8093 / 2 = 4046.5 (the remainder is 1, so 2 is not a divisor of 8093)
8093 / 3 = 2697.6666666667 (the remainder is 2, so 3 is not a divisor of 8093)
...
8093 / 8092 = 1.0001235788433 (the remainder is 1, so 8092 is not a divisor of 8093)
8093 / 8093 = 1 (the remainder is 0, so 8093 is a divisor of 8093)

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 8093 (i.e. 89.961102705558). 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 $\frac{N}{d} = D$ )

8093 / 1 = 8093 (the remainder is 0, so 1 and 8093 are divisors of 8093)
8093 / 2 = 4046.5 (the remainder is 1, so 2 is not a divisor of 8093)
8093 / 3 = 2697.6666666667 (the remainder is 2, so 3 is not a divisor of 8093)
...
8093 / 88 = 91.965909090909 (the remainder is 85, so 88 is not a divisor of 8093)
8093 / 89 = 90.932584269663 (the remainder is 83, so 89 is not a divisor of 8093)