What are the divisors of 2009?

1, 7, 41, 49, 287, 2009

There is a total of 6 positive divisors.
The sum of these divisors is 2394.
The arithmetic mean is 399.

6 odd divisors

1, 7, 41, 49, 287, 2009

How to compute the divisors of 2009?

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 2009 by each of the numbers from 1 to 2009 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)

2009 / 1 = 2009 (the remainder is 0, so 1 is a divisor of 2009)
2009 / 2 = 1004.5 (the remainder is 1, so 2 is not a divisor of 2009)
2009 / 3 = 669.66666666667 (the remainder is 2, so 3 is not a divisor of 2009)
...
2009 / 2008 = 1.0004980079681 (the remainder is 1, so 2008 is not a divisor of 2009)
2009 / 2009 = 1 (the remainder is 0, so 2009 is a divisor of 2009)

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 2009 (i.e. 44.82186966203). 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 $\frac{N}{d} = D$ )

2009 / 1 = 2009 (the remainder is 0, so 1 and 2009 are divisors of 2009)
2009 / 2 = 1004.5 (the remainder is 1, so 2 is not a divisor of 2009)
2009 / 3 = 669.66666666667 (the remainder is 2, so 3 is not a divisor of 2009)
...
2009 / 43 = 46.720930232558 (the remainder is 31, so 43 is not a divisor of 2009)
2009 / 44 = 45.659090909091 (the remainder is 29, so 44 is not a divisor of 2009)