What are the divisors of 8161?

1, 8161

There is a total of 2 positive divisors.
The sum of these divisors is 8162.
The arithmetic mean is 4081.

2 odd divisors

1, 8161

How to compute the divisors of 8161?

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 8161 by each of the numbers from 1 to 8161 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)

8161 / 1 = 8161 (the remainder is 0, so 1 is a divisor of 8161)
8161 / 2 = 4080.5 (the remainder is 1, so 2 is not a divisor of 8161)
8161 / 3 = 2720.3333333333 (the remainder is 1, so 3 is not a divisor of 8161)
...
8161 / 8160 = 1.0001225490196 (the remainder is 1, so 8160 is not a divisor of 8161)
8161 / 8161 = 1 (the remainder is 0, so 8161 is a divisor of 8161)

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 8161 (i.e. 90.338253248555). 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 $\frac{N}{d} = D$ )

8161 / 1 = 8161 (the remainder is 0, so 1 and 8161 are divisors of 8161)
8161 / 2 = 4080.5 (the remainder is 1, so 2 is not a divisor of 8161)
8161 / 3 = 2720.3333333333 (the remainder is 1, so 3 is not a divisor of 8161)
...
8161 / 89 = 91.696629213483 (the remainder is 62, so 89 is not a divisor of 8161)
8161 / 90 = 90.677777777778 (the remainder is 61, so 90 is not a divisor of 8161)