What are the divisors of 8221?

1, 8221

There is a total of 2 positive divisors.
The sum of these divisors is 8222.
The arithmetic mean is 4111.

2 odd divisors

1, 8221

How to compute the divisors of 8221?

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 8221 by each of the numbers from 1 to 8221 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)

8221 / 1 = 8221 (the remainder is 0, so 1 is a divisor of 8221)
8221 / 2 = 4110.5 (the remainder is 1, so 2 is not a divisor of 8221)
8221 / 3 = 2740.3333333333 (the remainder is 1, so 3 is not a divisor of 8221)
...
8221 / 8220 = 1.0001216545012 (the remainder is 1, so 8220 is not a divisor of 8221)
8221 / 8221 = 1 (the remainder is 0, so 8221 is a divisor of 8221)

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 8221 (i.e. 90.669730340395). 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 $\frac{N}{d} = D$ )

8221 / 1 = 8221 (the remainder is 0, so 1 and 8221 are divisors of 8221)
8221 / 2 = 4110.5 (the remainder is 1, so 2 is not a divisor of 8221)
8221 / 3 = 2740.3333333333 (the remainder is 1, so 3 is not a divisor of 8221)
...
8221 / 89 = 92.370786516854 (the remainder is 33, so 89 is not a divisor of 8221)
8221 / 90 = 91.344444444444 (the remainder is 31, so 90 is not a divisor of 8221)