What are the divisors of 8003?

1, 53, 151, 8003

There is a total of 4 positive divisors.
The sum of these divisors is 8208.
The arithmetic mean is 2052.

4 odd divisors

1, 53, 151, 8003

How to compute the divisors of 8003?

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 8003 by each of the numbers from 1 to 8003 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)

8003 / 1 = 8003 (the remainder is 0, so 1 is a divisor of 8003)
8003 / 2 = 4001.5 (the remainder is 1, so 2 is not a divisor of 8003)
8003 / 3 = 2667.6666666667 (the remainder is 2, so 3 is not a divisor of 8003)
...
8003 / 8002 = 1.0001249687578 (the remainder is 1, so 8002 is not a divisor of 8003)
8003 / 8003 = 1 (the remainder is 0, so 8003 is a divisor of 8003)

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 8003 (i.e. 89.459488037882). 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 $\frac{N}{d} = D$ )

8003 / 1 = 8003 (the remainder is 0, so 1 and 8003 are divisors of 8003)
8003 / 2 = 4001.5 (the remainder is 1, so 2 is not a divisor of 8003)
8003 / 3 = 2667.6666666667 (the remainder is 2, so 3 is not a divisor of 8003)
...
8003 / 88 = 90.943181818182 (the remainder is 83, so 88 is not a divisor of 8003)
8003 / 89 = 89.921348314607 (the remainder is 82, so 89 is not a divisor of 8003)