What are the divisors of 8113?

1, 7, 19, 61, 133, 427, 1159, 8113

There is a total of 8 positive divisors.
The sum of these divisors is 9920.
The arithmetic mean is 1240.

8 odd divisors

1, 7, 19, 61, 133, 427, 1159, 8113

How to compute the divisors of 8113?

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 8113 by each of the numbers from 1 to 8113 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)

8113 / 1 = 8113 (the remainder is 0, so 1 is a divisor of 8113)
8113 / 2 = 4056.5 (the remainder is 1, so 2 is not a divisor of 8113)
8113 / 3 = 2704.3333333333 (the remainder is 1, so 3 is not a divisor of 8113)
...
8113 / 8112 = 1.0001232741617 (the remainder is 1, so 8112 is not a divisor of 8113)
8113 / 8113 = 1 (the remainder is 0, so 8113 is a divisor of 8113)

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 8113 (i.e. 90.072193267401). 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 $\frac{N}{d} = D$ )

8113 / 1 = 8113 (the remainder is 0, so 1 and 8113 are divisors of 8113)
8113 / 2 = 4056.5 (the remainder is 1, so 2 is not a divisor of 8113)
8113 / 3 = 2704.3333333333 (the remainder is 1, so 3 is not a divisor of 8113)
...
8113 / 89 = 91.157303370787 (the remainder is 14, so 89 is not a divisor of 8113)
8113 / 90 = 90.144444444444 (the remainder is 13, so 90 is not a divisor of 8113)