What are the divisors of 408?

1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408

There is a total of 16 positive divisors.
The sum of these divisors is 1080.
The arithmetic mean is 67.5.

12 even divisors

2, 4, 6, 8, 12, 24, 34, 68, 102, 136, 204, 408

4 odd divisors

1, 3, 17, 51

How to compute the divisors of 408?

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 408 by each of the numbers from 1 to 408 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)

408 / 1 = 408 (the remainder is 0, so 1 is a divisor of 408)
408 / 2 = 204 (the remainder is 0, so 2 is a divisor of 408)
408 / 3 = 136 (the remainder is 0, so 3 is a divisor of 408)
...
408 / 407 = 1.002457002457 (the remainder is 1, so 407 is not a divisor of 408)
408 / 408 = 1 (the remainder is 0, so 408 is a divisor of 408)

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 408 (i.e. 20.199009876724). 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 $\frac{N}{d} = D$ )

408 / 1 = 408 (the remainder is 0, so 1 and 408 are divisors of 408)
408 / 2 = 204 (the remainder is 0, so 2 and 204 are divisors of 408)
408 / 3 = 136 (the remainder is 0, so 3 and 136 are divisors of 408)
...
408 / 19 = 21.473684210526 (the remainder is 9, so 19 is not a divisor of 408)
408 / 20 = 20.4 (the remainder is 8, so 20 is not a divisor of 408)