What are the divisors of 4065?

1, 3, 5, 15, 271, 813, 1355, 4065

There is a total of 8 positive divisors.
The sum of these divisors is 6528.
The arithmetic mean is 816.

8 odd divisors

1, 3, 5, 15, 271, 813, 1355, 4065

How to compute the divisors of 4065?

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 4065 by each of the numbers from 1 to 4065 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)

4065 / 1 = 4065 (the remainder is 0, so 1 is a divisor of 4065)
4065 / 2 = 2032.5 (the remainder is 1, so 2 is not a divisor of 4065)
4065 / 3 = 1355 (the remainder is 0, so 3 is a divisor of 4065)
...
4065 / 4064 = 1.0002460629921 (the remainder is 1, so 4064 is not a divisor of 4065)
4065 / 4065 = 1 (the remainder is 0, so 4065 is a divisor of 4065)

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 4065 (i.e. 63.75735251718). 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 $\frac{N}{d} = D$ )

4065 / 1 = 4065 (the remainder is 0, so 1 and 4065 are divisors of 4065)
4065 / 2 = 2032.5 (the remainder is 1, so 2 is not a divisor of 4065)
4065 / 3 = 1355 (the remainder is 0, so 3 and 1355 are divisors of 4065)
...
4065 / 62 = 65.564516129032 (the remainder is 35, so 62 is not a divisor of 4065)
4065 / 63 = 64.52380952381 (the remainder is 33, so 63 is not a divisor of 4065)