What are the divisors of 9097?

1, 11, 827, 9097

There is a total of 4 positive divisors.
The sum of these divisors is 9936.
The arithmetic mean is 2484.

4 odd divisors

1, 11, 827, 9097

How to compute the divisors of 9097?

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 9097 by each of the numbers from 1 to 9097 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)

9097 / 1 = 9097 (the remainder is 0, so 1 is a divisor of 9097)
9097 / 2 = 4548.5 (the remainder is 1, so 2 is not a divisor of 9097)
9097 / 3 = 3032.3333333333 (the remainder is 1, so 3 is not a divisor of 9097)
...
9097 / 9096 = 1.0001099384345 (the remainder is 1, so 9096 is not a divisor of 9097)
9097 / 9097 = 1 (the remainder is 0, so 9097 is a divisor of 9097)

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 9097 (i.e. 95.378194572974). 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 $\frac{N}{d} = D$ )

9097 / 1 = 9097 (the remainder is 0, so 1 and 9097 are divisors of 9097)
9097 / 2 = 4548.5 (the remainder is 1, so 2 is not a divisor of 9097)
9097 / 3 = 3032.3333333333 (the remainder is 1, so 3 is not a divisor of 9097)
...
9097 / 94 = 96.776595744681 (the remainder is 73, so 94 is not a divisor of 9097)
9097 / 95 = 95.757894736842 (the remainder is 72, so 95 is not a divisor of 9097)