What are the divisors of 2091?

1, 3, 17, 41, 51, 123, 697, 2091

There is a total of 8 positive divisors.
The sum of these divisors is 3024.
The arithmetic mean is 378.

8 odd divisors

1, 3, 17, 41, 51, 123, 697, 2091

How to compute the divisors of 2091?

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 2091 by each of the numbers from 1 to 2091 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)

2091 / 1 = 2091 (the remainder is 0, so 1 is a divisor of 2091)
2091 / 2 = 1045.5 (the remainder is 1, so 2 is not a divisor of 2091)
2091 / 3 = 697 (the remainder is 0, so 3 is a divisor of 2091)
...
2091 / 2090 = 1.0004784688995 (the remainder is 1, so 2090 is not a divisor of 2091)
2091 / 2091 = 1 (the remainder is 0, so 2091 is a divisor of 2091)

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 2091 (i.e. 45.727453460695). 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 $\frac{N}{d} = D$ )

2091 / 1 = 2091 (the remainder is 0, so 1 and 2091 are divisors of 2091)
2091 / 2 = 1045.5 (the remainder is 1, so 2 is not a divisor of 2091)
2091 / 3 = 697 (the remainder is 0, so 3 and 697 are divisors of 2091)
...
2091 / 44 = 47.522727272727 (the remainder is 23, so 44 is not a divisor of 2091)
2091 / 45 = 46.466666666667 (the remainder is 21, so 45 is not a divisor of 2091)