What are the divisors of 969?

1, 3, 17, 19, 51, 57, 323, 969

There is a total of 8 positive divisors.
The sum of these divisors is 1440.
The arithmetic mean is 180.

8 odd divisors

1, 3, 17, 19, 51, 57, 323, 969

How to compute the divisors of 969?

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 969 by each of the numbers from 1 to 969 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)

969 / 1 = 969 (the remainder is 0, so 1 is a divisor of 969)
969 / 2 = 484.5 (the remainder is 1, so 2 is not a divisor of 969)
969 / 3 = 323 (the remainder is 0, so 3 is a divisor of 969)
...
969 / 968 = 1.0010330578512 (the remainder is 1, so 968 is not a divisor of 969)
969 / 969 = 1 (the remainder is 0, so 969 is a divisor of 969)

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 969 (i.e. 31.128764832547). 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 $\frac{N}{d} = D$ )

969 / 1 = 969 (the remainder is 0, so 1 and 969 are divisors of 969)
969 / 2 = 484.5 (the remainder is 1, so 2 is not a divisor of 969)
969 / 3 = 323 (the remainder is 0, so 3 and 323 are divisors of 969)
...
969 / 30 = 32.3 (the remainder is 9, so 30 is not a divisor of 969)
969 / 31 = 31.258064516129 (the remainder is 8, so 31 is not a divisor of 969)