What are the divisors of 8107?

1, 11, 67, 121, 737, 8107

There is a total of 6 positive divisors.
The sum of these divisors is 9044.
The arithmetic mean is 1507.3333333333.

6 odd divisors

1, 11, 67, 121, 737, 8107

How to compute the divisors of 8107?

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 8107 by each of the numbers from 1 to 8107 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)

8107 / 1 = 8107 (the remainder is 0, so 1 is a divisor of 8107)
8107 / 2 = 4053.5 (the remainder is 1, so 2 is not a divisor of 8107)
8107 / 3 = 2702.3333333333 (the remainder is 1, so 3 is not a divisor of 8107)
...
8107 / 8106 = 1.0001233654083 (the remainder is 1, so 8106 is not a divisor of 8107)
8107 / 8107 = 1 (the remainder is 0, so 8107 is a divisor of 8107)

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 8107 (i.e. 90.038880490597). 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 $\frac{N}{d} = D$ )

8107 / 1 = 8107 (the remainder is 0, so 1 and 8107 are divisors of 8107)
8107 / 2 = 4053.5 (the remainder is 1, so 2 is not a divisor of 8107)
8107 / 3 = 2702.3333333333 (the remainder is 1, so 3 is not a divisor of 8107)
...
8107 / 89 = 91.089887640449 (the remainder is 8, so 89 is not a divisor of 8107)
8107 / 90 = 90.077777777778 (the remainder is 7, so 90 is not a divisor of 8107)