What are the divisors of 412?

1, 2, 4, 103, 206, 412

There is a total of 6 positive divisors.
The sum of these divisors is 728.
The arithmetic mean is 121.33333333333.

4 even divisors

2, 4, 206, 412

2 odd divisors

1, 103

How to compute the divisors of 412?

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 412 by each of the numbers from 1 to 412 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)

412 / 1 = 412 (the remainder is 0, so 1 is a divisor of 412)
412 / 2 = 206 (the remainder is 0, so 2 is a divisor of 412)
412 / 3 = 137.33333333333 (the remainder is 1, so 3 is not a divisor of 412)
...
412 / 411 = 1.0024330900243 (the remainder is 1, so 411 is not a divisor of 412)
412 / 412 = 1 (the remainder is 0, so 412 is a divisor of 412)

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 412 (i.e. 20.297783130184). 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 $\frac{N}{d} = D$ )

412 / 1 = 412 (the remainder is 0, so 1 and 412 are divisors of 412)
412 / 2 = 206 (the remainder is 0, so 2 and 206 are divisors of 412)
412 / 3 = 137.33333333333 (the remainder is 1, so 3 is not a divisor of 412)
...
412 / 19 = 21.684210526316 (the remainder is 13, so 19 is not a divisor of 412)
412 / 20 = 20.6 (the remainder is 12, so 20 is not a divisor of 412)