What are the divisors of 2081?

1, 2081

There is a total of 2 positive divisors.
The sum of these divisors is 2082.
The arithmetic mean is 1041.

2 odd divisors

1, 2081

How to compute the divisors of 2081?

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 2081 by each of the numbers from 1 to 2081 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)

2081 / 1 = 2081 (the remainder is 0, so 1 is a divisor of 2081)
2081 / 2 = 1040.5 (the remainder is 1, so 2 is not a divisor of 2081)
2081 / 3 = 693.66666666667 (the remainder is 2, so 3 is not a divisor of 2081)
...
2081 / 2080 = 1.0004807692308 (the remainder is 1, so 2080 is not a divisor of 2081)
2081 / 2081 = 1 (the remainder is 0, so 2081 is a divisor of 2081)

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 2081 (i.e. 45.617978911828). 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 $\frac{N}{d} = D$ )

2081 / 1 = 2081 (the remainder is 0, so 1 and 2081 are divisors of 2081)
2081 / 2 = 1040.5 (the remainder is 1, so 2 is not a divisor of 2081)
2081 / 3 = 693.66666666667 (the remainder is 2, so 3 is not a divisor of 2081)
...
2081 / 44 = 47.295454545455 (the remainder is 13, so 44 is not a divisor of 2081)
2081 / 45 = 46.244444444444 (the remainder is 11, so 45 is not a divisor of 2081)