What are the divisors of 8241?

1, 3, 41, 67, 123, 201, 2747, 8241

There is a total of 8 positive divisors.
The sum of these divisors is 11424.
The arithmetic mean is 1428.

8 odd divisors

1, 3, 41, 67, 123, 201, 2747, 8241

How to compute the divisors of 8241?

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 8241 by each of the numbers from 1 to 8241 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)

8241 / 1 = 8241 (the remainder is 0, so 1 is a divisor of 8241)
8241 / 2 = 4120.5 (the remainder is 1, so 2 is not a divisor of 8241)
8241 / 3 = 2747 (the remainder is 0, so 3 is a divisor of 8241)
...
8241 / 8240 = 1.0001213592233 (the remainder is 1, so 8240 is not a divisor of 8241)
8241 / 8241 = 1 (the remainder is 0, so 8241 is a divisor of 8241)

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 8241 (i.e. 90.779953734291). 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 $\frac{N}{d} = D$ )

8241 / 1 = 8241 (the remainder is 0, so 1 and 8241 are divisors of 8241)
8241 / 2 = 4120.5 (the remainder is 1, so 2 is not a divisor of 8241)
8241 / 3 = 2747 (the remainder is 0, so 3 and 2747 are divisors of 8241)
...
8241 / 89 = 92.595505617978 (the remainder is 53, so 89 is not a divisor of 8241)
8241 / 90 = 91.566666666667 (the remainder is 51, so 90 is not a divisor of 8241)