What are the divisors of 426?

1, 2, 3, 6, 71, 142, 213, 426

There is a total of 8 positive divisors.
The sum of these divisors is 864.
The arithmetic mean is 108.

4 even divisors

2, 6, 142, 426

4 odd divisors

1, 3, 71, 213

How to compute the divisors of 426?

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 426 by each of the numbers from 1 to 426 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)

426 / 1 = 426 (the remainder is 0, so 1 is a divisor of 426)
426 / 2 = 213 (the remainder is 0, so 2 is a divisor of 426)
426 / 3 = 142 (the remainder is 0, so 3 is a divisor of 426)
...
426 / 425 = 1.0023529411765 (the remainder is 1, so 425 is not a divisor of 426)
426 / 426 = 1 (the remainder is 0, so 426 is a divisor of 426)

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 426 (i.e. 20.63976744055). 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 $\frac{N}{d} = D$ )

426 / 1 = 426 (the remainder is 0, so 1 and 426 are divisors of 426)
426 / 2 = 213 (the remainder is 0, so 2 and 213 are divisors of 426)
426 / 3 = 142 (the remainder is 0, so 3 and 142 are divisors of 426)
...
426 / 19 = 22.421052631579 (the remainder is 8, so 19 is not a divisor of 426)
426 / 20 = 21.3 (the remainder is 6, so 20 is not a divisor of 426)