What are the divisors of 8177?

1, 13, 17, 37, 221, 481, 629, 8177

There is a total of 8 positive divisors.
The sum of these divisors is 9576.
The arithmetic mean is 1197.

8 odd divisors

1, 13, 17, 37, 221, 481, 629, 8177

How to compute the divisors of 8177?

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 8177 by each of the numbers from 1 to 8177 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)

8177 / 1 = 8177 (the remainder is 0, so 1 is a divisor of 8177)
8177 / 2 = 4088.5 (the remainder is 1, so 2 is not a divisor of 8177)
8177 / 3 = 2725.6666666667 (the remainder is 2, so 3 is not a divisor of 8177)
...
8177 / 8176 = 1.0001223091977 (the remainder is 1, so 8176 is not a divisor of 8177)
8177 / 8177 = 1 (the remainder is 0, so 8177 is a divisor of 8177)

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 8177 (i.e. 90.426765949026). 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 $\frac{N}{d} = D$ )

8177 / 1 = 8177 (the remainder is 0, so 1 and 8177 are divisors of 8177)
8177 / 2 = 4088.5 (the remainder is 1, so 2 is not a divisor of 8177)
8177 / 3 = 2725.6666666667 (the remainder is 2, so 3 is not a divisor of 8177)
...
8177 / 89 = 91.876404494382 (the remainder is 78, so 89 is not a divisor of 8177)
8177 / 90 = 90.855555555556 (the remainder is 77, so 90 is not a divisor of 8177)