What are the divisors of 8072?

1, 2, 4, 8, 1009, 2018, 4036, 8072

There is a total of 8 positive divisors.
The sum of these divisors is 15150.
The arithmetic mean is 1893.75.

6 even divisors

2, 4, 8, 2018, 4036, 8072

2 odd divisors

1, 1009

How to compute the divisors of 8072?

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 8072 by each of the numbers from 1 to 8072 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)

8072 / 1 = 8072 (the remainder is 0, so 1 is a divisor of 8072)
8072 / 2 = 4036 (the remainder is 0, so 2 is a divisor of 8072)
8072 / 3 = 2690.6666666667 (the remainder is 2, so 3 is not a divisor of 8072)
...
8072 / 8071 = 1.0001239003841 (the remainder is 1, so 8071 is not a divisor of 8072)
8072 / 8072 = 1 (the remainder is 0, so 8072 is a divisor of 8072)

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 8072 (i.e. 89.844309780865). 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 $\frac{N}{d} = D$ )

8072 / 1 = 8072 (the remainder is 0, so 1 and 8072 are divisors of 8072)
8072 / 2 = 4036 (the remainder is 0, so 2 and 4036 are divisors of 8072)
8072 / 3 = 2690.6666666667 (the remainder is 2, so 3 is not a divisor of 8072)
...
8072 / 88 = 91.727272727273 (the remainder is 64, so 88 is not a divisor of 8072)
8072 / 89 = 90.696629213483 (the remainder is 62, so 89 is not a divisor of 8072)