What are the divisors of 222?

1, 2, 3, 6, 37, 74, 111, 222

There is a total of 8 positive divisors.
The sum of these divisors is 456.
The arithmetic mean is 57.

4 even divisors

2, 6, 74, 222

4 odd divisors

1, 3, 37, 111

How to compute the divisors of 222?

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 222 by each of the numbers from 1 to 222 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)

222 / 1 = 222 (the remainder is 0, so 1 is a divisor of 222)
222 / 2 = 111 (the remainder is 0, so 2 is a divisor of 222)
222 / 3 = 74 (the remainder is 0, so 3 is a divisor of 222)
...
222 / 221 = 1.0045248868778 (the remainder is 1, so 221 is not a divisor of 222)
222 / 222 = 1 (the remainder is 0, so 222 is a divisor of 222)

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 222 (i.e. 14.899664425751). 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 $\frac{N}{d} = D$ )

222 / 1 = 222 (the remainder is 0, so 1 and 222 are divisors of 222)
222 / 2 = 111 (the remainder is 0, so 2 and 111 are divisors of 222)
222 / 3 = 74 (the remainder is 0, so 3 and 74 are divisors of 222)
...
222 / 13 = 17.076923076923 (the remainder is 1, so 13 is not a divisor of 222)
222 / 14 = 15.857142857143 (the remainder is 12, so 14 is not a divisor of 222)