What are the divisors of 9091?

1, 9091

There is a total of 2 positive divisors.
The sum of these divisors is 9092.
The arithmetic mean is 4546.

2 odd divisors

1, 9091

How to compute the divisors of 9091?

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 9091 by each of the numbers from 1 to 9091 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)

9091 / 1 = 9091 (the remainder is 0, so 1 is a divisor of 9091)
9091 / 2 = 4545.5 (the remainder is 1, so 2 is not a divisor of 9091)
9091 / 3 = 3030.3333333333 (the remainder is 1, so 3 is not a divisor of 9091)
...
9091 / 9090 = 1.0001100110011 (the remainder is 1, so 9090 is not a divisor of 9091)
9091 / 9091 = 1 (the remainder is 0, so 9091 is a divisor of 9091)

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 9091 (i.e. 95.346735654662). 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 $\frac{N}{d} = D$ )

9091 / 1 = 9091 (the remainder is 0, so 1 and 9091 are divisors of 9091)
9091 / 2 = 4545.5 (the remainder is 1, so 2 is not a divisor of 9091)
9091 / 3 = 3030.3333333333 (the remainder is 1, so 3 is not a divisor of 9091)
...
9091 / 94 = 96.712765957447 (the remainder is 67, so 94 is not a divisor of 9091)
9091 / 95 = 95.694736842105 (the remainder is 66, so 95 is not a divisor of 9091)