What are the divisors of 2011?

1, 2011

There is a total of 2 positive divisors.
The sum of these divisors is 2012.
The arithmetic mean is 1006.

2 odd divisors

1, 2011

How to compute the divisors of 2011?

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 2011 by each of the numbers from 1 to 2011 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)

2011 / 1 = 2011 (the remainder is 0, so 1 is a divisor of 2011)
2011 / 2 = 1005.5 (the remainder is 1, so 2 is not a divisor of 2011)
2011 / 3 = 670.33333333333 (the remainder is 1, so 3 is not a divisor of 2011)
...
2011 / 2010 = 1.0004975124378 (the remainder is 1, so 2010 is not a divisor of 2011)
2011 / 2011 = 1 (the remainder is 0, so 2011 is a divisor of 2011)

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 2011 (i.e. 44.844174649557). 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 $\frac{N}{d} = D$ )

2011 / 1 = 2011 (the remainder is 0, so 1 and 2011 are divisors of 2011)
2011 / 2 = 1005.5 (the remainder is 1, so 2 is not a divisor of 2011)
2011 / 3 = 670.33333333333 (the remainder is 1, so 3 is not a divisor of 2011)
...
2011 / 43 = 46.767441860465 (the remainder is 33, so 43 is not a divisor of 2011)
2011 / 44 = 45.704545454545 (the remainder is 31, so 44 is not a divisor of 2011)