What are the divisors of 438?

1, 2, 3, 6, 73, 146, 219, 438

There is a total of 8 positive divisors.
The sum of these divisors is 888.
The arithmetic mean is 111.

4 even divisors

2, 6, 146, 438

4 odd divisors

1, 3, 73, 219

How to compute the divisors of 438?

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 438 by each of the numbers from 1 to 438 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)

438 / 1 = 438 (the remainder is 0, so 1 is a divisor of 438)
438 / 2 = 219 (the remainder is 0, so 2 is a divisor of 438)
438 / 3 = 146 (the remainder is 0, so 3 is a divisor of 438)
...
438 / 437 = 1.0022883295195 (the remainder is 1, so 437 is not a divisor of 438)
438 / 438 = 1 (the remainder is 0, so 438 is a divisor of 438)

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 438 (i.e. 20.928449536456). 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 $\frac{N}{d} = D$ )

438 / 1 = 438 (the remainder is 0, so 1 and 438 are divisors of 438)
438 / 2 = 219 (the remainder is 0, so 2 and 219 are divisors of 438)
438 / 3 = 146 (the remainder is 0, so 3 and 146 are divisors of 438)
...
438 / 19 = 23.052631578947 (the remainder is 1, so 19 is not a divisor of 438)
438 / 20 = 21.9 (the remainder is 18, so 20 is not a divisor of 438)