What are the divisors of 8255?

1, 5, 13, 65, 127, 635, 1651, 8255

There is a total of 8 positive divisors.
The sum of these divisors is 10752.
The arithmetic mean is 1344.

8 odd divisors

1, 5, 13, 65, 127, 635, 1651, 8255

How to compute the divisors of 8255?

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 8255 by each of the numbers from 1 to 8255 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)

8255 / 1 = 8255 (the remainder is 0, so 1 is a divisor of 8255)
8255 / 2 = 4127.5 (the remainder is 1, so 2 is not a divisor of 8255)
8255 / 3 = 2751.6666666667 (the remainder is 2, so 3 is not a divisor of 8255)
...
8255 / 8254 = 1.0001211533802 (the remainder is 1, so 8254 is not a divisor of 8255)
8255 / 8255 = 1 (the remainder is 0, so 8255 is a divisor of 8255)

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 8255 (i.e. 90.857030547999). 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 $\frac{N}{d} = D$ )

8255 / 1 = 8255 (the remainder is 0, so 1 and 8255 are divisors of 8255)
8255 / 2 = 4127.5 (the remainder is 1, so 2 is not a divisor of 8255)
8255 / 3 = 2751.6666666667 (the remainder is 2, so 3 is not a divisor of 8255)
...
8255 / 89 = 92.752808988764 (the remainder is 67, so 89 is not a divisor of 8255)
8255 / 90 = 91.722222222222 (the remainder is 65, so 90 is not a divisor of 8255)