What are the divisors of 7009?

1, 43, 163, 7009

There is a total of 4 positive divisors.
The sum of these divisors is 7216.
The arithmetic mean is 1804.

4 odd divisors

1, 43, 163, 7009

How to compute the divisors of 7009?

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 7009 by each of the numbers from 1 to 7009 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)

7009 / 1 = 7009 (the remainder is 0, so 1 is a divisor of 7009)
7009 / 2 = 3504.5 (the remainder is 1, so 2 is not a divisor of 7009)
7009 / 3 = 2336.3333333333 (the remainder is 1, so 3 is not a divisor of 7009)
...
7009 / 7008 = 1.0001426940639 (the remainder is 1, so 7008 is not a divisor of 7009)
7009 / 7009 = 1 (the remainder is 0, so 7009 is a divisor of 7009)

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 7009 (i.e. 83.719770663804). 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 $\frac{N}{d} = D$ )

7009 / 1 = 7009 (the remainder is 0, so 1 and 7009 are divisors of 7009)
7009 / 2 = 3504.5 (the remainder is 1, so 2 is not a divisor of 7009)
7009 / 3 = 2336.3333333333 (the remainder is 1, so 3 is not a divisor of 7009)
...
7009 / 82 = 85.475609756098 (the remainder is 39, so 82 is not a divisor of 7009)
7009 / 83 = 84.44578313253 (the remainder is 37, so 83 is not a divisor of 7009)