What are the divisors of 9089?

1, 61, 149, 9089

There is a total of 4 positive divisors.
The sum of these divisors is 9300.
The arithmetic mean is 2325.

4 odd divisors

1, 61, 149, 9089

How to compute the divisors of 9089?

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 9089 by each of the numbers from 1 to 9089 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)

9089 / 1 = 9089 (the remainder is 0, so 1 is a divisor of 9089)
9089 / 2 = 4544.5 (the remainder is 1, so 2 is not a divisor of 9089)
9089 / 3 = 3029.6666666667 (the remainder is 2, so 3 is not a divisor of 9089)
...
9089 / 9088 = 1.0001100352113 (the remainder is 1, so 9088 is not a divisor of 9089)
9089 / 9089 = 1 (the remainder is 0, so 9089 is a divisor of 9089)

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 9089 (i.e. 95.336247041721). 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 $\frac{N}{d} = D$ )

9089 / 1 = 9089 (the remainder is 0, so 1 and 9089 are divisors of 9089)
9089 / 2 = 4544.5 (the remainder is 1, so 2 is not a divisor of 9089)
9089 / 3 = 3029.6666666667 (the remainder is 2, so 3 is not a divisor of 9089)
...
9089 / 94 = 96.691489361702 (the remainder is 65, so 94 is not a divisor of 9089)
9089 / 95 = 95.673684210526 (the remainder is 64, so 95 is not a divisor of 9089)