What are the divisors of 2163?

1, 3, 7, 21, 103, 309, 721, 2163

There is a total of 8 positive divisors.
The sum of these divisors is 3328.
The arithmetic mean is 416.

8 odd divisors

1, 3, 7, 21, 103, 309, 721, 2163

How to compute the divisors of 2163?

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 2163 by each of the numbers from 1 to 2163 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)

2163 / 1 = 2163 (the remainder is 0, so 1 is a divisor of 2163)
2163 / 2 = 1081.5 (the remainder is 1, so 2 is not a divisor of 2163)
2163 / 3 = 721 (the remainder is 0, so 3 is a divisor of 2163)
...
2163 / 2162 = 1.0004625346901 (the remainder is 1, so 2162 is not a divisor of 2163)
2163 / 2163 = 1 (the remainder is 0, so 2163 is a divisor of 2163)

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 2163 (i.e. 46.508063816934). 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 $\frac{N}{d} = D$ )

2163 / 1 = 2163 (the remainder is 0, so 1 and 2163 are divisors of 2163)
2163 / 2 = 1081.5 (the remainder is 1, so 2 is not a divisor of 2163)
2163 / 3 = 721 (the remainder is 0, so 3 and 721 are divisors of 2163)
...
2163 / 45 = 48.066666666667 (the remainder is 3, so 45 is not a divisor of 2163)
2163 / 46 = 47.021739130435 (the remainder is 1, so 46 is not a divisor of 2163)