What are the divisors of 963?

1, 3, 9, 107, 321, 963

There is a total of 6 positive divisors.
The sum of these divisors is 1404.
The arithmetic mean is 234.

6 odd divisors

1, 3, 9, 107, 321, 963

How to compute the divisors of 963?

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 963 by each of the numbers from 1 to 963 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)

963 / 1 = 963 (the remainder is 0, so 1 is a divisor of 963)
963 / 2 = 481.5 (the remainder is 1, so 2 is not a divisor of 963)
963 / 3 = 321 (the remainder is 0, so 3 is a divisor of 963)
...
963 / 962 = 1.0010395010395 (the remainder is 1, so 962 is not a divisor of 963)
963 / 963 = 1 (the remainder is 0, so 963 is a divisor of 963)

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 963 (i.e. 31.032241298366). 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 $\frac{N}{d} = D$ )

963 / 1 = 963 (the remainder is 0, so 1 and 963 are divisors of 963)
963 / 2 = 481.5 (the remainder is 1, so 2 is not a divisor of 963)
963 / 3 = 321 (the remainder is 0, so 3 and 321 are divisors of 963)
...
963 / 30 = 32.1 (the remainder is 3, so 30 is not a divisor of 963)
963 / 31 = 31.064516129032 (the remainder is 2, so 31 is not a divisor of 963)