What are the divisors of 8131?

1, 47, 173, 8131

There is a total of 4 positive divisors.
The sum of these divisors is 8352.
The arithmetic mean is 2088.

4 odd divisors

1, 47, 173, 8131

How to compute the divisors of 8131?

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 8131 by each of the numbers from 1 to 8131 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)

8131 / 1 = 8131 (the remainder is 0, so 1 is a divisor of 8131)
8131 / 2 = 4065.5 (the remainder is 1, so 2 is not a divisor of 8131)
8131 / 3 = 2710.3333333333 (the remainder is 1, so 3 is not a divisor of 8131)
...
8131 / 8130 = 1.00012300123 (the remainder is 1, so 8130 is not a divisor of 8131)
8131 / 8131 = 1 (the remainder is 0, so 8131 is a divisor of 8131)

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 8131 (i.e. 90.172057756269). 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 $\frac{N}{d} = D$ )

8131 / 1 = 8131 (the remainder is 0, so 1 and 8131 are divisors of 8131)
8131 / 2 = 4065.5 (the remainder is 1, so 2 is not a divisor of 8131)
8131 / 3 = 2710.3333333333 (the remainder is 1, so 3 is not a divisor of 8131)
...
8131 / 89 = 91.359550561798 (the remainder is 32, so 89 is not a divisor of 8131)
8131 / 90 = 90.344444444444 (the remainder is 31, so 90 is not a divisor of 8131)