What are the divisors of 424?

1, 2, 4, 8, 53, 106, 212, 424

There is a total of 8 positive divisors.
The sum of these divisors is 810.
The arithmetic mean is 101.25.

6 even divisors

2, 4, 8, 106, 212, 424

2 odd divisors

1, 53

How to compute the divisors of 424?

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 424 by each of the numbers from 1 to 424 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)

424 / 1 = 424 (the remainder is 0, so 1 is a divisor of 424)
424 / 2 = 212 (the remainder is 0, so 2 is a divisor of 424)
424 / 3 = 141.33333333333 (the remainder is 1, so 3 is not a divisor of 424)
...
424 / 423 = 1.0023640661939 (the remainder is 1, so 423 is not a divisor of 424)
424 / 424 = 1 (the remainder is 0, so 424 is a divisor of 424)

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 424 (i.e. 20.591260281974). 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 $\frac{N}{d} = D$ )

424 / 1 = 424 (the remainder is 0, so 1 and 424 are divisors of 424)
424 / 2 = 212 (the remainder is 0, so 2 and 212 are divisors of 424)
424 / 3 = 141.33333333333 (the remainder is 1, so 3 is not a divisor of 424)
...
424 / 19 = 22.315789473684 (the remainder is 6, so 19 is not a divisor of 424)
424 / 20 = 21.2 (the remainder is 4, so 20 is not a divisor of 424)