What are the divisors of 404?

1, 2, 4, 101, 202, 404

There is a total of 6 positive divisors.
The sum of these divisors is 714.
The arithmetic mean is 119.

4 even divisors

2, 4, 202, 404

2 odd divisors

1, 101

How to compute the divisors of 404?

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 404 by each of the numbers from 1 to 404 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)

404 / 1 = 404 (the remainder is 0, so 1 is a divisor of 404)
404 / 2 = 202 (the remainder is 0, so 2 is a divisor of 404)
404 / 3 = 134.66666666667 (the remainder is 2, so 3 is not a divisor of 404)
...
404 / 403 = 1.0024813895782 (the remainder is 1, so 403 is not a divisor of 404)
404 / 404 = 1 (the remainder is 0, so 404 is a divisor of 404)

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 404 (i.e. 20.099751242242). 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 $\frac{N}{d} = D$ )

404 / 1 = 404 (the remainder is 0, so 1 and 404 are divisors of 404)
404 / 2 = 202 (the remainder is 0, so 2 and 202 are divisors of 404)
404 / 3 = 134.66666666667 (the remainder is 2, so 3 is not a divisor of 404)
...
404 / 19 = 21.263157894737 (the remainder is 5, so 19 is not a divisor of 404)
404 / 20 = 20.2 (the remainder is 4, so 20 is not a divisor of 404)