What are the divisors of 8137?

1, 79, 103, 8137

There is a total of 4 positive divisors.
The sum of these divisors is 8320.
The arithmetic mean is 2080.

4 odd divisors

1, 79, 103, 8137

How to compute the divisors of 8137?

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 8137 by each of the numbers from 1 to 8137 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)

8137 / 1 = 8137 (the remainder is 0, so 1 is a divisor of 8137)
8137 / 2 = 4068.5 (the remainder is 1, so 2 is not a divisor of 8137)
8137 / 3 = 2712.3333333333 (the remainder is 1, so 3 is not a divisor of 8137)
...
8137 / 8136 = 1.0001229105211 (the remainder is 1, so 8136 is not a divisor of 8137)
8137 / 8137 = 1 (the remainder is 0, so 8137 is a divisor of 8137)

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 8137 (i.e. 90.205321350794). 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 $\frac{N}{d} = D$ )

8137 / 1 = 8137 (the remainder is 0, so 1 and 8137 are divisors of 8137)
8137 / 2 = 4068.5 (the remainder is 1, so 2 is not a divisor of 8137)
8137 / 3 = 2712.3333333333 (the remainder is 1, so 3 is not a divisor of 8137)
...
8137 / 89 = 91.426966292135 (the remainder is 38, so 89 is not a divisor of 8137)
8137 / 90 = 90.411111111111 (the remainder is 37, so 90 is not a divisor of 8137)