What are the divisors of 8121?

1, 3, 2707, 8121

There is a total of 4 positive divisors.
The sum of these divisors is 10832.
The arithmetic mean is 2708.

4 odd divisors

1, 3, 2707, 8121

How to compute the divisors of 8121?

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 8121 by each of the numbers from 1 to 8121 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)

8121 / 1 = 8121 (the remainder is 0, so 1 is a divisor of 8121)
8121 / 2 = 4060.5 (the remainder is 1, so 2 is not a divisor of 8121)
8121 / 3 = 2707 (the remainder is 0, so 3 is a divisor of 8121)
...
8121 / 8120 = 1.0001231527094 (the remainder is 1, so 8120 is not a divisor of 8121)
8121 / 8121 = 1 (the remainder is 0, so 8121 is a divisor of 8121)

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 8121 (i.e. 90.116591147247). 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 $\frac{N}{d} = D$ )

8121 / 1 = 8121 (the remainder is 0, so 1 and 8121 are divisors of 8121)
8121 / 2 = 4060.5 (the remainder is 1, so 2 is not a divisor of 8121)
8121 / 3 = 2707 (the remainder is 0, so 3 and 2707 are divisors of 8121)
...
8121 / 89 = 91.247191011236 (the remainder is 22, so 89 is not a divisor of 8121)
8121 / 90 = 90.233333333333 (the remainder is 21, so 90 is not a divisor of 8121)