What are the divisors of 1965?

1, 3, 5, 15, 131, 393, 655, 1965

There is a total of 8 positive divisors.
The sum of these divisors is 3168.
The arithmetic mean is 396.

8 odd divisors

1, 3, 5, 15, 131, 393, 655, 1965

How to compute the divisors of 1965?

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 1965 by each of the numbers from 1 to 1965 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)

1965 / 1 = 1965 (the remainder is 0, so 1 is a divisor of 1965)
1965 / 2 = 982.5 (the remainder is 1, so 2 is not a divisor of 1965)
1965 / 3 = 655 (the remainder is 0, so 3 is a divisor of 1965)
...
1965 / 1964 = 1.0005091649695 (the remainder is 1, so 1964 is not a divisor of 1965)
1965 / 1965 = 1 (the remainder is 0, so 1965 is a divisor of 1965)

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 1965 (i.e. 44.328320518603). 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 $\frac{N}{d} = D$ )

1965 / 1 = 1965 (the remainder is 0, so 1 and 1965 are divisors of 1965)
1965 / 2 = 982.5 (the remainder is 1, so 2 is not a divisor of 1965)
1965 / 3 = 655 (the remainder is 0, so 3 and 655 are divisors of 1965)
...
1965 / 43 = 45.697674418605 (the remainder is 30, so 43 is not a divisor of 1965)
1965 / 44 = 44.659090909091 (the remainder is 29, so 44 is not a divisor of 1965)