What are the divisors of 965?

1, 5, 193, 965

There is a total of 4 positive divisors.
The sum of these divisors is 1164.
The arithmetic mean is 291.

4 odd divisors

1, 5, 193, 965

How to compute the divisors of 965?

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 965 by each of the numbers from 1 to 965 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)

965 / 1 = 965 (the remainder is 0, so 1 is a divisor of 965)
965 / 2 = 482.5 (the remainder is 1, so 2 is not a divisor of 965)
965 / 3 = 321.66666666667 (the remainder is 2, so 3 is not a divisor of 965)
...
965 / 964 = 1.0010373443983 (the remainder is 1, so 964 is not a divisor of 965)
965 / 965 = 1 (the remainder is 0, so 965 is a divisor of 965)

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 965 (i.e. 31.064449134018). 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 $\frac{N}{d} = D$ )

965 / 1 = 965 (the remainder is 0, so 1 and 965 are divisors of 965)
965 / 2 = 482.5 (the remainder is 1, so 2 is not a divisor of 965)
965 / 3 = 321.66666666667 (the remainder is 2, so 3 is not a divisor of 965)
...
965 / 30 = 32.166666666667 (the remainder is 5, so 30 is not a divisor of 965)
965 / 31 = 31.129032258065 (the remainder is 4, so 31 is not a divisor of 965)