What are the divisors of 1015?

1, 5, 7, 29, 35, 145, 203, 1015

There is a total of 8 positive divisors.
The sum of these divisors is 1440.
The arithmetic mean is 180.

8 odd divisors

1, 5, 7, 29, 35, 145, 203, 1015

How to compute the divisors of 1015?

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 1015 by each of the numbers from 1 to 1015 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)

1015 / 1 = 1015 (the remainder is 0, so 1 is a divisor of 1015)
1015 / 2 = 507.5 (the remainder is 1, so 2 is not a divisor of 1015)
1015 / 3 = 338.33333333333 (the remainder is 1, so 3 is not a divisor of 1015)
...
1015 / 1014 = 1.0009861932939 (the remainder is 1, so 1014 is not a divisor of 1015)
1015 / 1015 = 1 (the remainder is 0, so 1015 is a divisor of 1015)

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 1015 (i.e. 31.859064644148). 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 $\frac{N}{d} = D$ )

1015 / 1 = 1015 (the remainder is 0, so 1 and 1015 are divisors of 1015)
1015 / 2 = 507.5 (the remainder is 1, so 2 is not a divisor of 1015)
1015 / 3 = 338.33333333333 (the remainder is 1, so 3 is not a divisor of 1015)
...
1015 / 30 = 33.833333333333 (the remainder is 25, so 30 is not a divisor of 1015)
1015 / 31 = 32.741935483871 (the remainder is 23, so 31 is not a divisor of 1015)