What are the divisors of 1065?

1, 3, 5, 15, 71, 213, 355, 1065

There is a total of 8 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 216.

8 odd divisors

1, 3, 5, 15, 71, 213, 355, 1065

How to compute the divisors of 1065?

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 1065 by each of the numbers from 1 to 1065 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)

1065 / 1 = 1065 (the remainder is 0, so 1 is a divisor of 1065)
1065 / 2 = 532.5 (the remainder is 1, so 2 is not a divisor of 1065)
1065 / 3 = 355 (the remainder is 0, so 3 is a divisor of 1065)
...
1065 / 1064 = 1.0009398496241 (the remainder is 1, so 1064 is not a divisor of 1065)
1065 / 1065 = 1 (the remainder is 0, so 1065 is a divisor of 1065)

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 1065 (i.e. 32.634337744161). 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 $\frac{N}{d} = D$ )

1065 / 1 = 1065 (the remainder is 0, so 1 and 1065 are divisors of 1065)
1065 / 2 = 532.5 (the remainder is 1, so 2 is not a divisor of 1065)
1065 / 3 = 355 (the remainder is 0, so 3 and 355 are divisors of 1065)
...
1065 / 31 = 34.354838709677 (the remainder is 11, so 31 is not a divisor of 1065)
1065 / 32 = 33.28125 (the remainder is 9, so 32 is not a divisor of 1065)