What are the divisors of 2089?

1, 2089

There is a total of 2 positive divisors.
The sum of these divisors is 2090.
The arithmetic mean is 1045.

2 odd divisors

1, 2089

How to compute the divisors of 2089?

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 2089 by each of the numbers from 1 to 2089 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)

2089 / 1 = 2089 (the remainder is 0, so 1 is a divisor of 2089)
2089 / 2 = 1044.5 (the remainder is 1, so 2 is not a divisor of 2089)
2089 / 3 = 696.33333333333 (the remainder is 1, so 3 is not a divisor of 2089)
...
2089 / 2088 = 1.0004789272031 (the remainder is 1, so 2088 is not a divisor of 2089)
2089 / 2089 = 1 (the remainder is 0, so 2089 is a divisor of 2089)

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 2089 (i.e. 45.705579528106). 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 $\frac{N}{d} = D$ )

2089 / 1 = 2089 (the remainder is 0, so 1 and 2089 are divisors of 2089)
2089 / 2 = 1044.5 (the remainder is 1, so 2 is not a divisor of 2089)
2089 / 3 = 696.33333333333 (the remainder is 1, so 3 is not a divisor of 2089)
...
2089 / 44 = 47.477272727273 (the remainder is 21, so 44 is not a divisor of 2089)
2089 / 45 = 46.422222222222 (the remainder is 19, so 45 is not a divisor of 2089)