What are the divisors of 8114?

1, 2, 4057, 8114

There is a total of 4 positive divisors.
The sum of these divisors is 12174.
The arithmetic mean is 3043.5.

2 even divisors

2, 8114

2 odd divisors

1, 4057

How to compute the divisors of 8114?

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 8114 by each of the numbers from 1 to 8114 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)

8114 / 1 = 8114 (the remainder is 0, so 1 is a divisor of 8114)
8114 / 2 = 4057 (the remainder is 0, so 2 is a divisor of 8114)
8114 / 3 = 2704.6666666667 (the remainder is 2, so 3 is not a divisor of 8114)
...
8114 / 8113 = 1.0001232589671 (the remainder is 1, so 8113 is not a divisor of 8114)
8114 / 8114 = 1 (the remainder is 0, so 8114 is a divisor of 8114)

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 8114 (i.e. 90.077744199108). 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 $\frac{N}{d} = D$ )

8114 / 1 = 8114 (the remainder is 0, so 1 and 8114 are divisors of 8114)
8114 / 2 = 4057 (the remainder is 0, so 2 and 4057 are divisors of 8114)
8114 / 3 = 2704.6666666667 (the remainder is 2, so 3 is not a divisor of 8114)
...
8114 / 89 = 91.168539325843 (the remainder is 15, so 89 is not a divisor of 8114)
8114 / 90 = 90.155555555556 (the remainder is 14, so 90 is not a divisor of 8114)