What are the divisors of 418?

1, 2, 11, 19, 22, 38, 209, 418

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

4 even divisors

2, 22, 38, 418

4 odd divisors

1, 11, 19, 209

How to compute the divisors of 418?

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 418 by each of the numbers from 1 to 418 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)

418 / 1 = 418 (the remainder is 0, so 1 is a divisor of 418)
418 / 2 = 209 (the remainder is 0, so 2 is a divisor of 418)
418 / 3 = 139.33333333333 (the remainder is 1, so 3 is not a divisor of 418)
...
418 / 417 = 1.0023980815348 (the remainder is 1, so 417 is not a divisor of 418)
418 / 418 = 1 (the remainder is 0, so 418 is a divisor of 418)

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 418 (i.e. 20.445048300261). 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 $\frac{N}{d} = D$ )

418 / 1 = 418 (the remainder is 0, so 1 and 418 are divisors of 418)
418 / 2 = 209 (the remainder is 0, so 2 and 209 are divisors of 418)
418 / 3 = 139.33333333333 (the remainder is 1, so 3 is not a divisor of 418)
...
418 / 19 = 22 (the remainder is 0, so 19 and 22 are divisors of 418)
418 / 20 = 20.9 (the remainder is 18, so 20 is not a divisor of 418)