What are the divisors of 8209?

1, 8209

There is a total of 2 positive divisors.
The sum of these divisors is 8210.
The arithmetic mean is 4105.

2 odd divisors

1, 8209

How to compute the divisors of 8209?

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 8209 by each of the numbers from 1 to 8209 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)

8209 / 1 = 8209 (the remainder is 0, so 1 is a divisor of 8209)
8209 / 2 = 4104.5 (the remainder is 1, so 2 is not a divisor of 8209)
8209 / 3 = 2736.3333333333 (the remainder is 1, so 3 is not a divisor of 8209)
...
8209 / 8208 = 1.0001218323587 (the remainder is 1, so 8208 is not a divisor of 8209)
8209 / 8209 = 1 (the remainder is 0, so 8209 is a divisor of 8209)

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 8209 (i.e. 90.603531939986). 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 $\frac{N}{d} = D$ )

8209 / 1 = 8209 (the remainder is 0, so 1 and 8209 are divisors of 8209)
8209 / 2 = 4104.5 (the remainder is 1, so 2 is not a divisor of 8209)
8209 / 3 = 2736.3333333333 (the remainder is 1, so 3 is not a divisor of 8209)
...
8209 / 89 = 92.23595505618 (the remainder is 21, so 89 is not a divisor of 8209)
8209 / 90 = 91.211111111111 (the remainder is 19, so 90 is not a divisor of 8209)