What are the divisors of 7065?

1, 3, 5, 9, 15, 45, 157, 471, 785, 1413, 2355, 7065

There is a total of 12 positive divisors.
The sum of these divisors is 12324.
The arithmetic mean is 1027.

12 odd divisors

1, 3, 5, 9, 15, 45, 157, 471, 785, 1413, 2355, 7065

How to compute the divisors of 7065?

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 7065 by each of the numbers from 1 to 7065 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)

7065 / 1 = 7065 (the remainder is 0, so 1 is a divisor of 7065)
7065 / 2 = 3532.5 (the remainder is 1, so 2 is not a divisor of 7065)
7065 / 3 = 2355 (the remainder is 0, so 3 is a divisor of 7065)
...
7065 / 7064 = 1.0001415628539 (the remainder is 1, so 7064 is not a divisor of 7065)
7065 / 7065 = 1 (the remainder is 0, so 7065 is a divisor of 7065)

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 7065 (i.e. 84.053554356731). 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 $\frac{N}{d} = D$ )

7065 / 1 = 7065 (the remainder is 0, so 1 and 7065 are divisors of 7065)
7065 / 2 = 3532.5 (the remainder is 1, so 2 is not a divisor of 7065)
7065 / 3 = 2355 (the remainder is 0, so 3 and 2355 are divisors of 7065)
...
7065 / 83 = 85.120481927711 (the remainder is 10, so 83 is not a divisor of 7065)
7065 / 84 = 84.107142857143 (the remainder is 9, so 84 is not a divisor of 7065)