What are the divisors of 2012?

1, 2, 4, 503, 1006, 2012

There is a total of 6 positive divisors.
The sum of these divisors is 3528.
The arithmetic mean is 588.

4 even divisors

2, 4, 1006, 2012

2 odd divisors

1, 503

How to compute the divisors of 2012?

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 2012 by each of the numbers from 1 to 2012 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)

2012 / 1 = 2012 (the remainder is 0, so 1 is a divisor of 2012)
2012 / 2 = 1006 (the remainder is 0, so 2 is a divisor of 2012)
2012 / 3 = 670.66666666667 (the remainder is 2, so 3 is not a divisor of 2012)
...
2012 / 2011 = 1.0004972650423 (the remainder is 1, so 2011 is not a divisor of 2012)
2012 / 2012 = 1 (the remainder is 0, so 2012 is a divisor of 2012)

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 2012 (i.e. 44.855322984012). 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 $\frac{N}{d} = D$ )

2012 / 1 = 2012 (the remainder is 0, so 1 and 2012 are divisors of 2012)
2012 / 2 = 1006 (the remainder is 0, so 2 and 1006 are divisors of 2012)
2012 / 3 = 670.66666666667 (the remainder is 2, so 3 is not a divisor of 2012)
...
2012 / 43 = 46.790697674419 (the remainder is 34, so 43 is not a divisor of 2012)
2012 / 44 = 45.727272727273 (the remainder is 32, so 44 is not a divisor of 2012)