What are the divisors of 8193?

1, 3, 2731, 8193

There is a total of 4 positive divisors.
The sum of these divisors is 10928.
The arithmetic mean is 2732.

4 odd divisors

1, 3, 2731, 8193

How to compute the divisors of 8193?

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 8193 by each of the numbers from 1 to 8193 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)

8193 / 1 = 8193 (the remainder is 0, so 1 is a divisor of 8193)
8193 / 2 = 4096.5 (the remainder is 1, so 2 is not a divisor of 8193)
8193 / 3 = 2731 (the remainder is 0, so 3 is a divisor of 8193)
...
8193 / 8192 = 1.0001220703125 (the remainder is 1, so 8192 is not a divisor of 8193)
8193 / 8193 = 1 (the remainder is 0, so 8193 is a divisor of 8193)

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 8193 (i.e. 90.515192095029). 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 $\frac{N}{d} = D$ )

8193 / 1 = 8193 (the remainder is 0, so 1 and 8193 are divisors of 8193)
8193 / 2 = 4096.5 (the remainder is 1, so 2 is not a divisor of 8193)
8193 / 3 = 2731 (the remainder is 0, so 3 and 2731 are divisors of 8193)
...
8193 / 89 = 92.056179775281 (the remainder is 5, so 89 is not a divisor of 8193)
8193 / 90 = 91.033333333333 (the remainder is 3, so 90 is not a divisor of 8193)