What are the divisors of 8220?

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 137, 274, 411, 548, 685, 822, 1370, 1644, 2055, 2740, 4110, 8220

There is a total of 24 positive divisors.
The sum of these divisors is 23184.
The arithmetic mean is 966.

16 even divisors

2, 4, 6, 10, 12, 20, 30, 60, 274, 548, 822, 1370, 1644, 2740, 4110, 8220

8 odd divisors

1, 3, 5, 15, 137, 411, 685, 2055

How to compute the divisors of 8220?

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 8220 by each of the numbers from 1 to 8220 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)

8220 / 1 = 8220 (the remainder is 0, so 1 is a divisor of 8220)
8220 / 2 = 4110 (the remainder is 0, so 2 is a divisor of 8220)
8220 / 3 = 2740 (the remainder is 0, so 3 is a divisor of 8220)
...
8220 / 8219 = 1.0001216693028 (the remainder is 1, so 8219 is not a divisor of 8220)
8220 / 8220 = 1 (the remainder is 0, so 8220 is a divisor of 8220)

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 8220 (i.e. 90.664215653145). 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 $\frac{N}{d} = D$ )

8220 / 1 = 8220 (the remainder is 0, so 1 and 8220 are divisors of 8220)
8220 / 2 = 4110 (the remainder is 0, so 2 and 4110 are divisors of 8220)
8220 / 3 = 2740 (the remainder is 0, so 3 and 2740 are divisors of 8220)
...
8220 / 89 = 92.359550561798 (the remainder is 32, so 89 is not a divisor of 8220)
8220 / 90 = 91.333333333333 (the remainder is 30, so 90 is not a divisor of 8220)