What are the divisors of 8117?

1, 8117

There is a total of 2 positive divisors.
The sum of these divisors is 8118.
The arithmetic mean is 4059.

2 odd divisors

1, 8117

How to compute the divisors of 8117?

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 8117 by each of the numbers from 1 to 8117 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)

8117 / 1 = 8117 (the remainder is 0, so 1 is a divisor of 8117)
8117 / 2 = 4058.5 (the remainder is 1, so 2 is not a divisor of 8117)
8117 / 3 = 2705.6666666667 (the remainder is 2, so 3 is not a divisor of 8117)
...
8117 / 8116 = 1.0001232134056 (the remainder is 1, so 8116 is not a divisor of 8117)
8117 / 8117 = 1 (the remainder is 0, so 8117 is a divisor of 8117)

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 8117 (i.e. 90.094394942194). 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 $\frac{N}{d} = D$ )

8117 / 1 = 8117 (the remainder is 0, so 1 and 8117 are divisors of 8117)
8117 / 2 = 4058.5 (the remainder is 1, so 2 is not a divisor of 8117)
8117 / 3 = 2705.6666666667 (the remainder is 2, so 3 is not a divisor of 8117)
...
8117 / 89 = 91.202247191011 (the remainder is 18, so 89 is not a divisor of 8117)
8117 / 90 = 90.188888888889 (the remainder is 17, so 90 is not a divisor of 8117)