What are the divisors of 2015?

1, 5, 13, 31, 65, 155, 403, 2015

There is a total of 8 positive divisors.
The sum of these divisors is 2688.
The arithmetic mean is 336.

8 odd divisors

1, 5, 13, 31, 65, 155, 403, 2015

How to compute the divisors of 2015?

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 2015 by each of the numbers from 1 to 2015 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)

2015 / 1 = 2015 (the remainder is 0, so 1 is a divisor of 2015)
2015 / 2 = 1007.5 (the remainder is 1, so 2 is not a divisor of 2015)
2015 / 3 = 671.66666666667 (the remainder is 2, so 3 is not a divisor of 2015)
...
2015 / 2014 = 1.0004965243297 (the remainder is 1, so 2014 is not a divisor of 2015)
2015 / 2015 = 1 (the remainder is 0, so 2015 is a divisor of 2015)

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 2015 (i.e. 44.888751374927). 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 $\frac{N}{d} = D$ )

2015 / 1 = 2015 (the remainder is 0, so 1 and 2015 are divisors of 2015)
2015 / 2 = 1007.5 (the remainder is 1, so 2 is not a divisor of 2015)
2015 / 3 = 671.66666666667 (the remainder is 2, so 3 is not a divisor of 2015)
...
2015 / 43 = 46.860465116279 (the remainder is 37, so 43 is not a divisor of 2015)
2015 / 44 = 45.795454545455 (the remainder is 35, so 44 is not a divisor of 2015)