What are the divisors of 5005?

1, 5, 7, 11, 13, 35, 55, 65, 77, 91, 143, 385, 455, 715, 1001, 5005

There is a total of 16 positive divisors.
The sum of these divisors is 8064.
The arithmetic mean is 504.

16 odd divisors

1, 5, 7, 11, 13, 35, 55, 65, 77, 91, 143, 385, 455, 715, 1001, 5005

How to compute the divisors of 5005?

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 5005 by each of the numbers from 1 to 5005 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)

5005 / 1 = 5005 (the remainder is 0, so 1 is a divisor of 5005)
5005 / 2 = 2502.5 (the remainder is 1, so 2 is not a divisor of 5005)
5005 / 3 = 1668.3333333333 (the remainder is 1, so 3 is not a divisor of 5005)
...
5005 / 5004 = 1.0001998401279 (the remainder is 1, so 5004 is not a divisor of 5005)
5005 / 5005 = 1 (the remainder is 0, so 5005 is a divisor of 5005)

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 5005 (i.e. 70.746024623296). 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 $\frac{N}{d} = D$ )

5005 / 1 = 5005 (the remainder is 0, so 1 and 5005 are divisors of 5005)
5005 / 2 = 2502.5 (the remainder is 1, so 2 is not a divisor of 5005)
5005 / 3 = 1668.3333333333 (the remainder is 1, so 3 is not a divisor of 5005)
...
5005 / 69 = 72.536231884058 (the remainder is 37, so 69 is not a divisor of 5005)
5005 / 70 = 71.5 (the remainder is 35, so 70 is not a divisor of 5005)