What are the divisors of 306?

1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306

There is a total of 12 positive divisors.
The sum of these divisors is 702.
The arithmetic mean is 58.5.

6 even divisors

2, 6, 18, 34, 102, 306

6 odd divisors

1, 3, 9, 17, 51, 153

How to compute the divisors of 306?

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 306 by each of the numbers from 1 to 306 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)

306 / 1 = 306 (the remainder is 0, so 1 is a divisor of 306)
306 / 2 = 153 (the remainder is 0, so 2 is a divisor of 306)
306 / 3 = 102 (the remainder is 0, so 3 is a divisor of 306)
...
306 / 305 = 1.0032786885246 (the remainder is 1, so 305 is not a divisor of 306)
306 / 306 = 1 (the remainder is 0, so 306 is a divisor of 306)

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 306 (i.e. 17.492855684536). 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 $\frac{N}{d} = D$ )

306 / 1 = 306 (the remainder is 0, so 1 and 306 are divisors of 306)
306 / 2 = 153 (the remainder is 0, so 2 and 153 are divisors of 306)
306 / 3 = 102 (the remainder is 0, so 3 and 102 are divisors of 306)
...
306 / 16 = 19.125 (the remainder is 2, so 16 is not a divisor of 306)
306 / 17 = 18 (the remainder is 0, so 17 and 18 are divisors of 306)