What are the divisors of 8215?

1, 5, 31, 53, 155, 265, 1643, 8215

There is a total of 8 positive divisors.
The sum of these divisors is 10368.
The arithmetic mean is 1296.

8 odd divisors

1, 5, 31, 53, 155, 265, 1643, 8215

How to compute the divisors of 8215?

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 8215 by each of the numbers from 1 to 8215 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)

8215 / 1 = 8215 (the remainder is 0, so 1 is a divisor of 8215)
8215 / 2 = 4107.5 (the remainder is 1, so 2 is not a divisor of 8215)
8215 / 3 = 2738.3333333333 (the remainder is 1, so 3 is not a divisor of 8215)
...
8215 / 8214 = 1.000121743365 (the remainder is 1, so 8214 is not a divisor of 8215)
8215 / 8215 = 1 (the remainder is 0, so 8215 is a divisor of 8215)

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 8215 (i.e. 90.636637183867). 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 $\frac{N}{d} = D$ )

8215 / 1 = 8215 (the remainder is 0, so 1 and 8215 are divisors of 8215)
8215 / 2 = 4107.5 (the remainder is 1, so 2 is not a divisor of 8215)
8215 / 3 = 2738.3333333333 (the remainder is 1, so 3 is not a divisor of 8215)
...
8215 / 89 = 92.303370786517 (the remainder is 27, so 89 is not a divisor of 8215)
8215 / 90 = 91.277777777778 (the remainder is 25, so 90 is not a divisor of 8215)