What are the divisors of 660?

1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660

There is a total of 24 positive divisors.
The sum of these divisors is 2016.
The arithmetic mean is 84.

16 even divisors

2, 4, 6, 10, 12, 20, 22, 30, 44, 60, 66, 110, 132, 220, 330, 660

8 odd divisors

1, 3, 5, 11, 15, 33, 55, 165

How to compute the divisors of 660?

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 660 by each of the numbers from 1 to 660 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)

660 / 1 = 660 (the remainder is 0, so 1 is a divisor of 660)
660 / 2 = 330 (the remainder is 0, so 2 is a divisor of 660)
660 / 3 = 220 (the remainder is 0, so 3 is a divisor of 660)
...
660 / 659 = 1.0015174506829 (the remainder is 1, so 659 is not a divisor of 660)
660 / 660 = 1 (the remainder is 0, so 660 is a divisor of 660)

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 660 (i.e. 25.69046515733). 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 $\frac{N}{d} = D$ )

660 / 1 = 660 (the remainder is 0, so 1 and 660 are divisors of 660)
660 / 2 = 330 (the remainder is 0, so 2 and 330 are divisors of 660)
660 / 3 = 220 (the remainder is 0, so 3 and 220 are divisors of 660)
...
660 / 24 = 27.5 (the remainder is 12, so 24 is not a divisor of 660)
660 / 25 = 26.4 (the remainder is 10, so 25 is not a divisor of 660)