What are the divisors of 405?

1, 3, 5, 9, 15, 27, 45, 81, 135, 405

There is a total of 10 positive divisors.
The sum of these divisors is 726.
The arithmetic mean is 72.6.

10 odd divisors

1, 3, 5, 9, 15, 27, 45, 81, 135, 405

How to compute the divisors of 405?

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 405 by each of the numbers from 1 to 405 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)

405 / 1 = 405 (the remainder is 0, so 1 is a divisor of 405)
405 / 2 = 202.5 (the remainder is 1, so 2 is not a divisor of 405)
405 / 3 = 135 (the remainder is 0, so 3 is a divisor of 405)
...
405 / 404 = 1.0024752475248 (the remainder is 1, so 404 is not a divisor of 405)
405 / 405 = 1 (the remainder is 0, so 405 is a divisor of 405)

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 405 (i.e. 20.124611797498). 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 $\frac{N}{d} = D$ )

405 / 1 = 405 (the remainder is 0, so 1 and 405 are divisors of 405)
405 / 2 = 202.5 (the remainder is 1, so 2 is not a divisor of 405)
405 / 3 = 135 (the remainder is 0, so 3 and 135 are divisors of 405)
...
405 / 19 = 21.315789473684 (the remainder is 6, so 19 is not a divisor of 405)
405 / 20 = 20.25 (the remainder is 5, so 20 is not a divisor of 405)