What are the divisors of 8195?

1, 5, 11, 55, 149, 745, 1639, 8195

There is a total of 8 positive divisors.
The sum of these divisors is 10800.
The arithmetic mean is 1350.

8 odd divisors

1, 5, 11, 55, 149, 745, 1639, 8195

How to compute the divisors of 8195?

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 8195 by each of the numbers from 1 to 8195 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)

8195 / 1 = 8195 (the remainder is 0, so 1 is a divisor of 8195)
8195 / 2 = 4097.5 (the remainder is 1, so 2 is not a divisor of 8195)
8195 / 3 = 2731.6666666667 (the remainder is 2, so 3 is not a divisor of 8195)
...
8195 / 8194 = 1.0001220405175 (the remainder is 1, so 8194 is not a divisor of 8195)
8195 / 8195 = 1 (the remainder is 0, so 8195 is a divisor of 8195)

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 8195 (i.e. 90.526239290053). 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 $\frac{N}{d} = D$ )

8195 / 1 = 8195 (the remainder is 0, so 1 and 8195 are divisors of 8195)
8195 / 2 = 4097.5 (the remainder is 1, so 2 is not a divisor of 8195)
8195 / 3 = 2731.6666666667 (the remainder is 2, so 3 is not a divisor of 8195)
...
8195 / 89 = 92.078651685393 (the remainder is 7, so 89 is not a divisor of 8195)
8195 / 90 = 91.055555555556 (the remainder is 5, so 90 is not a divisor of 8195)