What are the divisors of 8305?

1, 5, 11, 55, 151, 755, 1661, 8305

There is a total of 8 positive divisors.
The sum of these divisors is 10944.
The arithmetic mean is 1368.

8 odd divisors

1, 5, 11, 55, 151, 755, 1661, 8305

How to compute the divisors of 8305?

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 8305 by each of the numbers from 1 to 8305 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)

8305 / 1 = 8305 (the remainder is 0, so 1 is a divisor of 8305)
8305 / 2 = 4152.5 (the remainder is 1, so 2 is not a divisor of 8305)
8305 / 3 = 2768.3333333333 (the remainder is 1, so 3 is not a divisor of 8305)
...
8305 / 8304 = 1.0001204238921 (the remainder is 1, so 8304 is not a divisor of 8305)
8305 / 8305 = 1 (the remainder is 0, so 8305 is a divisor of 8305)

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 8305 (i.e. 91.131772724994). 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 $\frac{N}{d} = D$ )

8305 / 1 = 8305 (the remainder is 0, so 1 and 8305 are divisors of 8305)
8305 / 2 = 4152.5 (the remainder is 1, so 2 is not a divisor of 8305)
8305 / 3 = 2768.3333333333 (the remainder is 1, so 3 is not a divisor of 8305)
...
8305 / 90 = 92.277777777778 (the remainder is 25, so 90 is not a divisor of 8305)
8305 / 91 = 91.263736263736 (the remainder is 24, so 91 is not a divisor of 8305)