What are the divisors of 5003?

1, 5003

There is a total of 2 positive divisors.
The sum of these divisors is 5004.
The arithmetic mean is 2502.

2 odd divisors

1, 5003

How to compute the divisors of 5003?

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 5003 by each of the numbers from 1 to 5003 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)

5003 / 1 = 5003 (the remainder is 0, so 1 is a divisor of 5003)
5003 / 2 = 2501.5 (the remainder is 1, so 2 is not a divisor of 5003)
5003 / 3 = 1667.6666666667 (the remainder is 2, so 3 is not a divisor of 5003)
...
5003 / 5002 = 1.000199920032 (the remainder is 1, so 5002 is not a divisor of 5003)
5003 / 5003 = 1 (the remainder is 0, so 5003 is a divisor of 5003)

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 5003 (i.e. 70.731888141064). 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 $\frac{N}{d} = D$ )

5003 / 1 = 5003 (the remainder is 0, so 1 and 5003 are divisors of 5003)
5003 / 2 = 2501.5 (the remainder is 1, so 2 is not a divisor of 5003)
5003 / 3 = 1667.6666666667 (the remainder is 2, so 3 is not a divisor of 5003)
...
5003 / 69 = 72.507246376812 (the remainder is 35, so 69 is not a divisor of 5003)
5003 / 70 = 71.471428571429 (the remainder is 33, so 70 is not a divisor of 5003)