What are the divisors of 2181?

1, 3, 727, 2181

There is a total of 4 positive divisors.
The sum of these divisors is 2912.
The arithmetic mean is 728.

4 odd divisors

1, 3, 727, 2181

How to compute the divisors of 2181?

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 2181 by each of the numbers from 1 to 2181 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)

2181 / 1 = 2181 (the remainder is 0, so 1 is a divisor of 2181)
2181 / 2 = 1090.5 (the remainder is 1, so 2 is not a divisor of 2181)
2181 / 3 = 727 (the remainder is 0, so 3 is a divisor of 2181)
...
2181 / 2180 = 1.0004587155963 (the remainder is 1, so 2180 is not a divisor of 2181)
2181 / 2181 = 1 (the remainder is 0, so 2181 is a divisor of 2181)

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 2181 (i.e. 46.701177715342). 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 $\frac{N}{d} = D$ )

2181 / 1 = 2181 (the remainder is 0, so 1 and 2181 are divisors of 2181)
2181 / 2 = 1090.5 (the remainder is 1, so 2 is not a divisor of 2181)
2181 / 3 = 727 (the remainder is 0, so 3 and 727 are divisors of 2181)
...
2181 / 45 = 48.466666666667 (the remainder is 21, so 45 is not a divisor of 2181)
2181 / 46 = 47.413043478261 (the remainder is 19, so 46 is not a divisor of 2181)