What are the divisors of 406?

1, 2, 7, 14, 29, 58, 203, 406

There is a total of 8 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 90.

4 even divisors

2, 14, 58, 406

4 odd divisors

1, 7, 29, 203

How to compute the divisors of 406?

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 406 by each of the numbers from 1 to 406 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)

406 / 1 = 406 (the remainder is 0, so 1 is a divisor of 406)
406 / 2 = 203 (the remainder is 0, so 2 is a divisor of 406)
406 / 3 = 135.33333333333 (the remainder is 1, so 3 is not a divisor of 406)
...
406 / 405 = 1.0024691358025 (the remainder is 1, so 405 is not a divisor of 406)
406 / 406 = 1 (the remainder is 0, so 406 is a divisor of 406)

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 406 (i.e. 20.14944167961). 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 $\frac{N}{d} = D$ )

406 / 1 = 406 (the remainder is 0, so 1 and 406 are divisors of 406)
406 / 2 = 203 (the remainder is 0, so 2 and 203 are divisors of 406)
406 / 3 = 135.33333333333 (the remainder is 1, so 3 is not a divisor of 406)
...
406 / 19 = 21.368421052632 (the remainder is 7, so 19 is not a divisor of 406)
406 / 20 = 20.3 (the remainder is 6, so 20 is not a divisor of 406)