What are the divisors of 8065?

1, 5, 1613, 8065

There is a total of 4 positive divisors.
The sum of these divisors is 9684.
The arithmetic mean is 2421.

4 odd divisors

1, 5, 1613, 8065

How to compute the divisors of 8065?

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 8065 by each of the numbers from 1 to 8065 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)

8065 / 1 = 8065 (the remainder is 0, so 1 is a divisor of 8065)
8065 / 2 = 4032.5 (the remainder is 1, so 2 is not a divisor of 8065)
8065 / 3 = 2688.3333333333 (the remainder is 1, so 3 is not a divisor of 8065)
...
8065 / 8064 = 1.0001240079365 (the remainder is 1, so 8064 is not a divisor of 8065)
8065 / 8065 = 1 (the remainder is 0, so 8065 is a divisor of 8065)

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 8065 (i.e. 89.805345052508). 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 $\frac{N}{d} = D$ )

8065 / 1 = 8065 (the remainder is 0, so 1 and 8065 are divisors of 8065)
8065 / 2 = 4032.5 (the remainder is 1, so 2 is not a divisor of 8065)
8065 / 3 = 2688.3333333333 (the remainder is 1, so 3 is not a divisor of 8065)
...
8065 / 88 = 91.647727272727 (the remainder is 57, so 88 is not a divisor of 8065)
8065 / 89 = 90.61797752809 (the remainder is 55, so 89 is not a divisor of 8065)