What are the divisors of 8251?

1, 37, 223, 8251

There is a total of 4 positive divisors.
The sum of these divisors is 8512.
The arithmetic mean is 2128.

4 odd divisors

1, 37, 223, 8251

How to compute the divisors of 8251?

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 8251 by each of the numbers from 1 to 8251 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)

8251 / 1 = 8251 (the remainder is 0, so 1 is a divisor of 8251)
8251 / 2 = 4125.5 (the remainder is 1, so 2 is not a divisor of 8251)
8251 / 3 = 2750.3333333333 (the remainder is 1, so 3 is not a divisor of 8251)
...
8251 / 8250 = 1.0001212121212 (the remainder is 1, so 8250 is not a divisor of 8251)
8251 / 8251 = 1 (the remainder is 0, so 8251 is a divisor of 8251)

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 8251 (i.e. 90.835015274948). 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 $\frac{N}{d} = D$ )

8251 / 1 = 8251 (the remainder is 0, so 1 and 8251 are divisors of 8251)
8251 / 2 = 4125.5 (the remainder is 1, so 2 is not a divisor of 8251)
8251 / 3 = 2750.3333333333 (the remainder is 1, so 3 is not a divisor of 8251)
...
8251 / 89 = 92.707865168539 (the remainder is 63, so 89 is not a divisor of 8251)
8251 / 90 = 91.677777777778 (the remainder is 61, so 90 is not a divisor of 8251)