What are the divisors of 1005?

1, 3, 5, 15, 67, 201, 335, 1005

There is a total of 8 positive divisors.
The sum of these divisors is 1632.
The arithmetic mean is 204.

8 odd divisors

1, 3, 5, 15, 67, 201, 335, 1005

How to compute the divisors of 1005?

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 1005 by each of the numbers from 1 to 1005 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)

1005 / 1 = 1005 (the remainder is 0, so 1 is a divisor of 1005)
1005 / 2 = 502.5 (the remainder is 1, so 2 is not a divisor of 1005)
1005 / 3 = 335 (the remainder is 0, so 3 is a divisor of 1005)
...
1005 / 1004 = 1.0009960159363 (the remainder is 1, so 1004 is not a divisor of 1005)
1005 / 1005 = 1 (the remainder is 0, so 1005 is a divisor of 1005)

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 1005 (i.e. 31.701734968295). 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 $\frac{N}{d} = D$ )

1005 / 1 = 1005 (the remainder is 0, so 1 and 1005 are divisors of 1005)
1005 / 2 = 502.5 (the remainder is 1, so 2 is not a divisor of 1005)
1005 / 3 = 335 (the remainder is 0, so 3 and 335 are divisors of 1005)
...
1005 / 30 = 33.5 (the remainder is 15, so 30 is not a divisor of 1005)
1005 / 31 = 32.41935483871 (the remainder is 13, so 31 is not a divisor of 1005)