What are the divisors of 8141?

1, 7, 1163, 8141

There is a total of 4 positive divisors.
The sum of these divisors is 9312.
The arithmetic mean is 2328.

4 odd divisors

1, 7, 1163, 8141

How to compute the divisors of 8141?

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 8141 by each of the numbers from 1 to 8141 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)

8141 / 1 = 8141 (the remainder is 0, so 1 is a divisor of 8141)
8141 / 2 = 4070.5 (the remainder is 1, so 2 is not a divisor of 8141)
8141 / 3 = 2713.6666666667 (the remainder is 2, so 3 is not a divisor of 8141)
...
8141 / 8140 = 1.0001228501229 (the remainder is 1, so 8140 is not a divisor of 8141)
8141 / 8141 = 1 (the remainder is 0, so 8141 is a divisor of 8141)

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 8141 (i.e. 90.227490267656). 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 $\frac{N}{d} = D$ )

8141 / 1 = 8141 (the remainder is 0, so 1 and 8141 are divisors of 8141)
8141 / 2 = 4070.5 (the remainder is 1, so 2 is not a divisor of 8141)
8141 / 3 = 2713.6666666667 (the remainder is 2, so 3 is not a divisor of 8141)
...
8141 / 89 = 91.47191011236 (the remainder is 42, so 89 is not a divisor of 8141)
8141 / 90 = 90.455555555556 (the remainder is 41, so 90 is not a divisor of 8141)