What are the divisors of 4015?

1, 5, 11, 55, 73, 365, 803, 4015

There is a total of 8 positive divisors.
The sum of these divisors is 5328.
The arithmetic mean is 666.

8 odd divisors

1, 5, 11, 55, 73, 365, 803, 4015

How to compute the divisors of 4015?

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 4015 by each of the numbers from 1 to 4015 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)

4015 / 1 = 4015 (the remainder is 0, so 1 is a divisor of 4015)
4015 / 2 = 2007.5 (the remainder is 1, so 2 is not a divisor of 4015)
4015 / 3 = 1338.3333333333 (the remainder is 1, so 3 is not a divisor of 4015)
...
4015 / 4014 = 1.0002491280518 (the remainder is 1, so 4014 is not a divisor of 4015)
4015 / 4015 = 1 (the remainder is 0, so 4015 is a divisor of 4015)

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 4015 (i.e. 63.364027649764). 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 $\frac{N}{d} = D$ )

4015 / 1 = 4015 (the remainder is 0, so 1 and 4015 are divisors of 4015)
4015 / 2 = 2007.5 (the remainder is 1, so 2 is not a divisor of 4015)
4015 / 3 = 1338.3333333333 (the remainder is 1, so 3 is not a divisor of 4015)
...
4015 / 62 = 64.758064516129 (the remainder is 47, so 62 is not a divisor of 4015)
4015 / 63 = 63.730158730159 (the remainder is 46, so 63 is not a divisor of 4015)