What are the divisors of 409?

1, 409

There is a total of 2 positive divisors.
The sum of these divisors is 410.
The arithmetic mean is 205.

2 odd divisors

1, 409

How to compute the divisors of 409?

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 409 by each of the numbers from 1 to 409 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)

409 / 1 = 409 (the remainder is 0, so 1 is a divisor of 409)
409 / 2 = 204.5 (the remainder is 1, so 2 is not a divisor of 409)
409 / 3 = 136.33333333333 (the remainder is 1, so 3 is not a divisor of 409)
...
409 / 408 = 1.0024509803922 (the remainder is 1, so 408 is not a divisor of 409)
409 / 409 = 1 (the remainder is 0, so 409 is a divisor of 409)

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 409 (i.e. 20.223748416157). 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 $\frac{N}{d} = D$ )

409 / 1 = 409 (the remainder is 0, so 1 and 409 are divisors of 409)
409 / 2 = 204.5 (the remainder is 1, so 2 is not a divisor of 409)
409 / 3 = 136.33333333333 (the remainder is 1, so 3 is not a divisor of 409)
...
409 / 19 = 21.526315789474 (the remainder is 10, so 19 is not a divisor of 409)
409 / 20 = 20.45 (the remainder is 9, so 20 is not a divisor of 409)