What are the divisors of 9109?

1, 9109

There is a total of 2 positive divisors.
The sum of these divisors is 9110.
The arithmetic mean is 4555.

2 odd divisors

1, 9109

How to compute the divisors of 9109?

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 9109 by each of the numbers from 1 to 9109 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)

9109 / 1 = 9109 (the remainder is 0, so 1 is a divisor of 9109)
9109 / 2 = 4554.5 (the remainder is 1, so 2 is not a divisor of 9109)
9109 / 3 = 3036.3333333333 (the remainder is 1, so 3 is not a divisor of 9109)
...
9109 / 9108 = 1.0001097935881 (the remainder is 1, so 9108 is not a divisor of 9109)
9109 / 9109 = 1 (the remainder is 0, so 9109 is a divisor of 9109)

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 9109 (i.e. 95.441081301502). 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 $\frac{N}{d} = D$ )

9109 / 1 = 9109 (the remainder is 0, so 1 and 9109 are divisors of 9109)
9109 / 2 = 4554.5 (the remainder is 1, so 2 is not a divisor of 9109)
9109 / 3 = 3036.3333333333 (the remainder is 1, so 3 is not a divisor of 9109)
...
9109 / 94 = 96.904255319149 (the remainder is 85, so 94 is not a divisor of 9109)
9109 / 95 = 95.884210526316 (the remainder is 84, so 95 is not a divisor of 9109)