What are the divisors of 8027?

1, 23, 349, 8027

There is a total of 4 positive divisors.
The sum of these divisors is 8400.
The arithmetic mean is 2100.

4 odd divisors

1, 23, 349, 8027

How to compute the divisors of 8027?

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 8027 by each of the numbers from 1 to 8027 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)

8027 / 1 = 8027 (the remainder is 0, so 1 is a divisor of 8027)
8027 / 2 = 4013.5 (the remainder is 1, so 2 is not a divisor of 8027)
8027 / 3 = 2675.6666666667 (the remainder is 2, so 3 is not a divisor of 8027)
...
8027 / 8026 = 1.000124595066 (the remainder is 1, so 8026 is not a divisor of 8027)
8027 / 8027 = 1 (the remainder is 0, so 8027 is a divisor of 8027)

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 8027 (i.e. 89.593526551866). 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 $\frac{N}{d} = D$ )

8027 / 1 = 8027 (the remainder is 0, so 1 and 8027 are divisors of 8027)
8027 / 2 = 4013.5 (the remainder is 1, so 2 is not a divisor of 8027)
8027 / 3 = 2675.6666666667 (the remainder is 2, so 3 is not a divisor of 8027)
...
8027 / 88 = 91.215909090909 (the remainder is 19, so 88 is not a divisor of 8027)
8027 / 89 = 90.191011235955 (the remainder is 17, so 89 is not a divisor of 8027)