What are the divisors of 1009?

1, 1009

There is a total of 2 positive divisors.
The sum of these divisors is 1010.
The arithmetic mean is 505.

2 odd divisors

1, 1009

How to compute the divisors of 1009?

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 1009 by each of the numbers from 1 to 1009 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)

1009 / 1 = 1009 (the remainder is 0, so 1 is a divisor of 1009)
1009 / 2 = 504.5 (the remainder is 1, so 2 is not a divisor of 1009)
1009 / 3 = 336.33333333333 (the remainder is 1, so 3 is not a divisor of 1009)
...
1009 / 1008 = 1.0009920634921 (the remainder is 1, so 1008 is not a divisor of 1009)
1009 / 1009 = 1 (the remainder is 0, so 1009 is a divisor of 1009)

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 1009 (i.e. 31.764760348537). 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 $\frac{N}{d} = D$ )

1009 / 1 = 1009 (the remainder is 0, so 1 and 1009 are divisors of 1009)
1009 / 2 = 504.5 (the remainder is 1, so 2 is not a divisor of 1009)
1009 / 3 = 336.33333333333 (the remainder is 1, so 3 is not a divisor of 1009)
...
1009 / 30 = 33.633333333333 (the remainder is 19, so 30 is not a divisor of 1009)
1009 / 31 = 32.548387096774 (the remainder is 17, so 31 is not a divisor of 1009)