What are the divisors of 8051?

1, 83, 97, 8051

There is a total of 4 positive divisors.
The sum of these divisors is 8232.
The arithmetic mean is 2058.

4 odd divisors

1, 83, 97, 8051

How to compute the divisors of 8051?

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 8051 by each of the numbers from 1 to 8051 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)

8051 / 1 = 8051 (the remainder is 0, so 1 is a divisor of 8051)
8051 / 2 = 4025.5 (the remainder is 1, so 2 is not a divisor of 8051)
8051 / 3 = 2683.6666666667 (the remainder is 2, so 3 is not a divisor of 8051)
...
8051 / 8050 = 1.0001242236025 (the remainder is 1, so 8050 is not a divisor of 8051)
8051 / 8051 = 1 (the remainder is 0, so 8051 is a divisor of 8051)

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 8051 (i.e. 89.727364833701). 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 $\frac{N}{d} = D$ )

8051 / 1 = 8051 (the remainder is 0, so 1 and 8051 are divisors of 8051)
8051 / 2 = 4025.5 (the remainder is 1, so 2 is not a divisor of 8051)
8051 / 3 = 2683.6666666667 (the remainder is 2, so 3 is not a divisor of 8051)
...
8051 / 88 = 91.488636363636 (the remainder is 43, so 88 is not a divisor of 8051)
8051 / 89 = 90.460674157303 (the remainder is 41, so 89 is not a divisor of 8051)