What are the divisors of 4003?

1, 4003

There is a total of 2 positive divisors.
The sum of these divisors is 4004.
The arithmetic mean is 2002.

2 odd divisors

1, 4003

How to compute the divisors of 4003?

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 4003 by each of the numbers from 1 to 4003 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)

4003 / 1 = 4003 (the remainder is 0, so 1 is a divisor of 4003)
4003 / 2 = 2001.5 (the remainder is 1, so 2 is not a divisor of 4003)
4003 / 3 = 1334.3333333333 (the remainder is 1, so 3 is not a divisor of 4003)
...
4003 / 4002 = 1.0002498750625 (the remainder is 1, so 4002 is not a divisor of 4003)
4003 / 4003 = 1 (the remainder is 0, so 4003 is a divisor of 4003)

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 4003 (i.e. 63.269265840533). 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 $\frac{N}{d} = D$ )

4003 / 1 = 4003 (the remainder is 0, so 1 and 4003 are divisors of 4003)
4003 / 2 = 2001.5 (the remainder is 1, so 2 is not a divisor of 4003)
4003 / 3 = 1334.3333333333 (the remainder is 1, so 3 is not a divisor of 4003)
...
4003 / 62 = 64.564516129032 (the remainder is 35, so 62 is not a divisor of 4003)
4003 / 63 = 63.539682539683 (the remainder is 34, so 63 is not a divisor of 4003)