What are the divisors of 966?

1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966

There is a total of 16 positive divisors.
The sum of these divisors is 2304.
The arithmetic mean is 144.

8 even divisors

2, 6, 14, 42, 46, 138, 322, 966

8 odd divisors

1, 3, 7, 21, 23, 69, 161, 483

How to compute the divisors of 966?

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 966 by each of the numbers from 1 to 966 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)

966 / 1 = 966 (the remainder is 0, so 1 is a divisor of 966)
966 / 2 = 483 (the remainder is 0, so 2 is a divisor of 966)
966 / 3 = 322 (the remainder is 0, so 3 is a divisor of 966)
...
966 / 965 = 1.0010362694301 (the remainder is 1, so 965 is not a divisor of 966)
966 / 966 = 1 (the remainder is 0, so 966 is a divisor of 966)

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 966 (i.e. 31.08054053584). 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 $\frac{N}{d} = D$ )

966 / 1 = 966 (the remainder is 0, so 1 and 966 are divisors of 966)
966 / 2 = 483 (the remainder is 0, so 2 and 483 are divisors of 966)
966 / 3 = 322 (the remainder is 0, so 3 and 322 are divisors of 966)
...
966 / 30 = 32.2 (the remainder is 6, so 30 is not a divisor of 966)
966 / 31 = 31.161290322581 (the remainder is 5, so 31 is not a divisor of 966)