What are the divisors of 422?

1, 2, 211, 422

There is a total of 4 positive divisors.
The sum of these divisors is 636.
The arithmetic mean is 159.

2 even divisors

2, 422

2 odd divisors

1, 211

How to compute the divisors of 422?

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 422 by each of the numbers from 1 to 422 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)

422 / 1 = 422 (the remainder is 0, so 1 is a divisor of 422)
422 / 2 = 211 (the remainder is 0, so 2 is a divisor of 422)
422 / 3 = 140.66666666667 (the remainder is 2, so 3 is not a divisor of 422)
...
422 / 421 = 1.0023752969121 (the remainder is 1, so 421 is not a divisor of 422)
422 / 422 = 1 (the remainder is 0, so 422 is a divisor of 422)

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 422 (i.e. 20.542638584174). 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 $\frac{N}{d} = D$ )

422 / 1 = 422 (the remainder is 0, so 1 and 422 are divisors of 422)
422 / 2 = 211 (the remainder is 0, so 2 and 211 are divisors of 422)
422 / 3 = 140.66666666667 (the remainder is 2, so 3 is not a divisor of 422)
...
422 / 19 = 22.210526315789 (the remainder is 4, so 19 is not a divisor of 422)
422 / 20 = 21.1 (the remainder is 2, so 20 is not a divisor of 422)