What are the divisors of 4009?

1, 19, 211, 4009

There is a total of 4 positive divisors.
The sum of these divisors is 4240.
The arithmetic mean is 1060.

4 odd divisors

1, 19, 211, 4009

How to compute the divisors of 4009?

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 4009 by each of the numbers from 1 to 4009 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)

4009 / 1 = 4009 (the remainder is 0, so 1 is a divisor of 4009)
4009 / 2 = 2004.5 (the remainder is 1, so 2 is not a divisor of 4009)
4009 / 3 = 1336.3333333333 (the remainder is 1, so 3 is not a divisor of 4009)
...
4009 / 4008 = 1.000249500998 (the remainder is 1, so 4008 is not a divisor of 4009)
4009 / 4009 = 1 (the remainder is 0, so 4009 is a divisor of 4009)

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 4009 (i.e. 63.316664473107). 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 $\frac{N}{d} = D$ )

4009 / 1 = 4009 (the remainder is 0, so 1 and 4009 are divisors of 4009)
4009 / 2 = 2004.5 (the remainder is 1, so 2 is not a divisor of 4009)
4009 / 3 = 1336.3333333333 (the remainder is 1, so 3 is not a divisor of 4009)
...
4009 / 62 = 64.661290322581 (the remainder is 41, so 62 is not a divisor of 4009)
4009 / 63 = 63.634920634921 (the remainder is 40, so 63 is not a divisor of 4009)