What are the divisors of 8171?

1, 8171

There is a total of 2 positive divisors.
The sum of these divisors is 8172.
The arithmetic mean is 4086.

2 odd divisors

1, 8171

How to compute the divisors of 8171?

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 8171 by each of the numbers from 1 to 8171 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)

8171 / 1 = 8171 (the remainder is 0, so 1 is a divisor of 8171)
8171 / 2 = 4085.5 (the remainder is 1, so 2 is not a divisor of 8171)
8171 / 3 = 2723.6666666667 (the remainder is 2, so 3 is not a divisor of 8171)
...
8171 / 8170 = 1.0001223990208 (the remainder is 1, so 8170 is not a divisor of 8171)
8171 / 8171 = 1 (the remainder is 0, so 8171 is a divisor of 8171)

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 8171 (i.e. 90.393583843102). 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 $\frac{N}{d} = D$ )

8171 / 1 = 8171 (the remainder is 0, so 1 and 8171 are divisors of 8171)
8171 / 2 = 4085.5 (the remainder is 1, so 2 is not a divisor of 8171)
8171 / 3 = 2723.6666666667 (the remainder is 2, so 3 is not a divisor of 8171)
...
8171 / 89 = 91.808988764045 (the remainder is 72, so 89 is not a divisor of 8171)
8171 / 90 = 90.788888888889 (the remainder is 71, so 90 is not a divisor of 8171)