What are the divisors of 198?

1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198

There is a total of 12 positive divisors.
The sum of these divisors is 468.
The arithmetic mean is 39.

6 even divisors

2, 6, 18, 22, 66, 198

6 odd divisors

1, 3, 9, 11, 33, 99

How to compute the divisors of 198?

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 198 by each of the numbers from 1 to 198 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)

198 / 1 = 198 (the remainder is 0, so 1 is a divisor of 198)
198 / 2 = 99 (the remainder is 0, so 2 is a divisor of 198)
198 / 3 = 66 (the remainder is 0, so 3 is a divisor of 198)
...
198 / 197 = 1.005076142132 (the remainder is 1, so 197 is not a divisor of 198)
198 / 198 = 1 (the remainder is 0, so 198 is a divisor of 198)

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 198 (i.e. 14.07124727947). 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 $\frac{N}{d} = D$ )

198 / 1 = 198 (the remainder is 0, so 1 and 198 are divisors of 198)
198 / 2 = 99 (the remainder is 0, so 2 and 99 are divisors of 198)
198 / 3 = 66 (the remainder is 0, so 3 and 66 are divisors of 198)
...
198 / 13 = 15.230769230769 (the remainder is 3, so 13 is not a divisor of 198)
198 / 14 = 14.142857142857 (the remainder is 2, so 14 is not a divisor of 198)