What are the divisors of 2016?

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016

There is a total of 36 positive divisors.
The sum of these divisors is 6552.
The arithmetic mean is 182.

30 even divisors

2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 42, 48, 56, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008, 2016

6 odd divisors

1, 3, 7, 9, 21, 63

How to compute the divisors of 2016?

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 2016 by each of the numbers from 1 to 2016 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)

2016 / 1 = 2016 (the remainder is 0, so 1 is a divisor of 2016)
2016 / 2 = 1008 (the remainder is 0, so 2 is a divisor of 2016)
2016 / 3 = 672 (the remainder is 0, so 3 is a divisor of 2016)
...
2016 / 2015 = 1.0004962779156 (the remainder is 1, so 2015 is not a divisor of 2016)
2016 / 2016 = 1 (the remainder is 0, so 2016 is a divisor of 2016)

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 2016 (i.e. 44.899888641287). 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 $\frac{N}{d} = D$ )

2016 / 1 = 2016 (the remainder is 0, so 1 and 2016 are divisors of 2016)
2016 / 2 = 1008 (the remainder is 0, so 2 and 1008 are divisors of 2016)
2016 / 3 = 672 (the remainder is 0, so 3 and 672 are divisors of 2016)
...
2016 / 43 = 46.883720930233 (the remainder is 38, so 43 is not a divisor of 2016)
2016 / 44 = 45.818181818182 (the remainder is 36, so 44 is not a divisor of 2016)