What are the divisors of 428?

1, 2, 4, 107, 214, 428

There is a total of 6 positive divisors.
The sum of these divisors is 756.
The arithmetic mean is 126.

4 even divisors

2, 4, 214, 428

2 odd divisors

1, 107

How to compute the divisors of 428?

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 428 by each of the numbers from 1 to 428 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)

428 / 1 = 428 (the remainder is 0, so 1 is a divisor of 428)
428 / 2 = 214 (the remainder is 0, so 2 is a divisor of 428)
428 / 3 = 142.66666666667 (the remainder is 2, so 3 is not a divisor of 428)
...
428 / 427 = 1.0023419203747 (the remainder is 1, so 427 is not a divisor of 428)
428 / 428 = 1 (the remainder is 0, so 428 is a divisor of 428)

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 428 (i.e. 20.688160865577). 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 $\frac{N}{d} = D$ )

428 / 1 = 428 (the remainder is 0, so 1 and 428 are divisors of 428)
428 / 2 = 214 (the remainder is 0, so 2 and 214 are divisors of 428)
428 / 3 = 142.66666666667 (the remainder is 2, so 3 is not a divisor of 428)
...
428 / 19 = 22.526315789474 (the remainder is 10, so 19 is not a divisor of 428)
428 / 20 = 21.4 (the remainder is 8, so 20 is not a divisor of 428)