What are the divisors of 444?

1, 2, 3, 4, 6, 12, 37, 74, 111, 148, 222, 444

There is a total of 12 positive divisors.
The sum of these divisors is 1064.
The arithmetic mean is 88.666666666667.

8 even divisors

2, 4, 6, 12, 74, 148, 222, 444

4 odd divisors

1, 3, 37, 111

How to compute the divisors of 444?

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 444 by each of the numbers from 1 to 444 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)

444 / 1 = 444 (the remainder is 0, so 1 is a divisor of 444)
444 / 2 = 222 (the remainder is 0, so 2 is a divisor of 444)
444 / 3 = 148 (the remainder is 0, so 3 is a divisor of 444)
...
444 / 443 = 1.0022573363431 (the remainder is 1, so 443 is not a divisor of 444)
444 / 444 = 1 (the remainder is 0, so 444 is a divisor of 444)

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 444 (i.e. 21.071307505705). 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 $\frac{N}{d} = D$ )

444 / 1 = 444 (the remainder is 0, so 1 and 444 are divisors of 444)
444 / 2 = 222 (the remainder is 0, so 2 and 222 are divisors of 444)
444 / 3 = 148 (the remainder is 0, so 3 and 148 are divisors of 444)
...
444 / 20 = 22.2 (the remainder is 4, so 20 is not a divisor of 444)
444 / 21 = 21.142857142857 (the remainder is 3, so 21 is not a divisor of 444)