What are the divisors of 8201?

1, 59, 139, 8201

There is a total of 4 positive divisors.
The sum of these divisors is 8400.
The arithmetic mean is 2100.

4 odd divisors

1, 59, 139, 8201

How to compute the divisors of 8201?

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 8201 by each of the numbers from 1 to 8201 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)

8201 / 1 = 8201 (the remainder is 0, so 1 is a divisor of 8201)
8201 / 2 = 4100.5 (the remainder is 1, so 2 is not a divisor of 8201)
8201 / 3 = 2733.6666666667 (the remainder is 2, so 3 is not a divisor of 8201)
...
8201 / 8200 = 1.0001219512195 (the remainder is 1, so 8200 is not a divisor of 8201)
8201 / 8201 = 1 (the remainder is 0, so 8201 is a divisor of 8201)

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 8201 (i.e. 90.559372789347). 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 $\frac{N}{d} = D$ )

8201 / 1 = 8201 (the remainder is 0, so 1 and 8201 are divisors of 8201)
8201 / 2 = 4100.5 (the remainder is 1, so 2 is not a divisor of 8201)
8201 / 3 = 2733.6666666667 (the remainder is 2, so 3 is not a divisor of 8201)
...
8201 / 89 = 92.14606741573 (the remainder is 13, so 89 is not a divisor of 8201)
8201 / 90 = 91.122222222222 (the remainder is 11, so 90 is not a divisor of 8201)