What are the divisors of 8135?

1, 5, 1627, 8135

There is a total of 4 positive divisors.
The sum of these divisors is 9768.
The arithmetic mean is 2442.

4 odd divisors

1, 5, 1627, 8135

How to compute the divisors of 8135?

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 8135 by each of the numbers from 1 to 8135 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)

8135 / 1 = 8135 (the remainder is 0, so 1 is a divisor of 8135)
8135 / 2 = 4067.5 (the remainder is 1, so 2 is not a divisor of 8135)
8135 / 3 = 2711.6666666667 (the remainder is 2, so 3 is not a divisor of 8135)
...
8135 / 8134 = 1.0001229407426 (the remainder is 1, so 8134 is not a divisor of 8135)
8135 / 8135 = 1 (the remainder is 0, so 8135 is a divisor of 8135)

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 8135 (i.e. 90.194234849019). 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 $\frac{N}{d} = D$ )

8135 / 1 = 8135 (the remainder is 0, so 1 and 8135 are divisors of 8135)
8135 / 2 = 4067.5 (the remainder is 1, so 2 is not a divisor of 8135)
8135 / 3 = 2711.6666666667 (the remainder is 2, so 3 is not a divisor of 8135)
...
8135 / 89 = 91.404494382022 (the remainder is 36, so 89 is not a divisor of 8135)
8135 / 90 = 90.388888888889 (the remainder is 35, so 90 is not a divisor of 8135)