What are the divisors of 8297?

1, 8297

There is a total of 2 positive divisors.
The sum of these divisors is 8298.
The arithmetic mean is 4149.

2 odd divisors

1, 8297

How to compute the divisors of 8297?

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 8297 by each of the numbers from 1 to 8297 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)

8297 / 1 = 8297 (the remainder is 0, so 1 is a divisor of 8297)
8297 / 2 = 4148.5 (the remainder is 1, so 2 is not a divisor of 8297)
8297 / 3 = 2765.6666666667 (the remainder is 2, so 3 is not a divisor of 8297)
...
8297 / 8296 = 1.0001205400193 (the remainder is 1, so 8296 is not a divisor of 8297)
8297 / 8297 = 1 (the remainder is 0, so 8297 is a divisor of 8297)

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 8297 (i.e. 91.087869664407). 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 $\frac{N}{d} = D$ )

8297 / 1 = 8297 (the remainder is 0, so 1 and 8297 are divisors of 8297)
8297 / 2 = 4148.5 (the remainder is 1, so 2 is not a divisor of 8297)
8297 / 3 = 2765.6666666667 (the remainder is 2, so 3 is not a divisor of 8297)
...
8297 / 90 = 92.188888888889 (the remainder is 17, so 90 is not a divisor of 8297)
8297 / 91 = 91.175824175824 (the remainder is 16, so 91 is not a divisor of 8297)