What are the divisors of 8270?

1, 2, 5, 10, 827, 1654, 4135, 8270

There is a total of 8 positive divisors.
The sum of these divisors is 14904.
The arithmetic mean is 1863.

4 even divisors

2, 10, 1654, 8270

4 odd divisors

1, 5, 827, 4135

How to compute the divisors of 8270?

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 8270 by each of the numbers from 1 to 8270 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)

8270 / 1 = 8270 (the remainder is 0, so 1 is a divisor of 8270)
8270 / 2 = 4135 (the remainder is 0, so 2 is a divisor of 8270)
8270 / 3 = 2756.6666666667 (the remainder is 2, so 3 is not a divisor of 8270)
...
8270 / 8269 = 1.0001209336074 (the remainder is 1, so 8269 is not a divisor of 8270)
8270 / 8270 = 1 (the remainder is 0, so 8270 is a divisor of 8270)

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 8270 (i.e. 90.939540355117). 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 $\frac{N}{d} = D$ )

8270 / 1 = 8270 (the remainder is 0, so 1 and 8270 are divisors of 8270)
8270 / 2 = 4135 (the remainder is 0, so 2 and 4135 are divisors of 8270)
8270 / 3 = 2756.6666666667 (the remainder is 2, so 3 is not a divisor of 8270)
...
8270 / 89 = 92.921348314607 (the remainder is 82, so 89 is not a divisor of 8270)
8270 / 90 = 91.888888888889 (the remainder is 80, so 90 is not a divisor of 8270)