What are the divisors of 2106?

1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 81, 117, 162, 234, 351, 702, 1053, 2106

There is a total of 20 positive divisors.
The sum of these divisors is 5082.
The arithmetic mean is 254.1.

10 even divisors

2, 6, 18, 26, 54, 78, 162, 234, 702, 2106

10 odd divisors

1, 3, 9, 13, 27, 39, 81, 117, 351, 1053

How to compute the divisors of 2106?

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 2106 by each of the numbers from 1 to 2106 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)

2106 / 1 = 2106 (the remainder is 0, so 1 is a divisor of 2106)
2106 / 2 = 1053 (the remainder is 0, so 2 is a divisor of 2106)
2106 / 3 = 702 (the remainder is 0, so 3 is a divisor of 2106)
...
2106 / 2105 = 1.0004750593824 (the remainder is 1, so 2105 is not a divisor of 2106)
2106 / 2106 = 1 (the remainder is 0, so 2106 is a divisor of 2106)

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 2106 (i.e. 45.891175622335). 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 $\frac{N}{d} = D$ )

2106 / 1 = 2106 (the remainder is 0, so 1 and 2106 are divisors of 2106)
2106 / 2 = 1053 (the remainder is 0, so 2 and 1053 are divisors of 2106)
2106 / 3 = 702 (the remainder is 0, so 3 and 702 are divisors of 2106)
...
2106 / 44 = 47.863636363636 (the remainder is 38, so 44 is not a divisor of 2106)
2106 / 45 = 46.8 (the remainder is 36, so 45 is not a divisor of 2106)