What are the divisors of 8078?

1, 2, 7, 14, 577, 1154, 4039, 8078

There is a total of 8 positive divisors.
The sum of these divisors is 13872.
The arithmetic mean is 1734.

4 even divisors

2, 14, 1154, 8078

4 odd divisors

1, 7, 577, 4039

How to compute the divisors of 8078?

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 8078 by each of the numbers from 1 to 8078 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)

8078 / 1 = 8078 (the remainder is 0, so 1 is a divisor of 8078)
8078 / 2 = 4039 (the remainder is 0, so 2 is a divisor of 8078)
8078 / 3 = 2692.6666666667 (the remainder is 2, so 3 is not a divisor of 8078)
...
8078 / 8077 = 1.0001238083447 (the remainder is 1, so 8077 is not a divisor of 8078)
8078 / 8078 = 1 (the remainder is 0, so 8078 is a divisor of 8078)

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 8078 (i.e. 89.877694674485). 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 $\frac{N}{d} = D$ )

8078 / 1 = 8078 (the remainder is 0, so 1 and 8078 are divisors of 8078)
8078 / 2 = 4039 (the remainder is 0, so 2 and 4039 are divisors of 8078)
8078 / 3 = 2692.6666666667 (the remainder is 2, so 3 is not a divisor of 8078)
...
8078 / 88 = 91.795454545455 (the remainder is 70, so 88 is not a divisor of 8078)
8078 / 89 = 90.76404494382 (the remainder is 68, so 89 is not a divisor of 8078)