What are the divisors of 1230?

1, 2, 3, 5, 6, 10, 15, 30, 41, 82, 123, 205, 246, 410, 615, 1230

There is a total of 16 positive divisors.
The sum of these divisors is 3024.
The arithmetic mean is 189.

8 even divisors

2, 6, 10, 30, 82, 246, 410, 1230

8 odd divisors

1, 3, 5, 15, 41, 123, 205, 615

How to compute the divisors of 1230?

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 1230 by each of the numbers from 1 to 1230 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)

1230 / 1 = 1230 (the remainder is 0, so 1 is a divisor of 1230)
1230 / 2 = 615 (the remainder is 0, so 2 is a divisor of 1230)
1230 / 3 = 410 (the remainder is 0, so 3 is a divisor of 1230)
...
1230 / 1229 = 1.0008136696501 (the remainder is 1, so 1229 is not a divisor of 1230)
1230 / 1230 = 1 (the remainder is 0, so 1230 is a divisor of 1230)

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 1230 (i.e. 35.0713558335). 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 $\frac{N}{d} = D$ )

1230 / 1 = 1230 (the remainder is 0, so 1 and 1230 are divisors of 1230)
1230 / 2 = 615 (the remainder is 0, so 2 and 615 are divisors of 1230)
1230 / 3 = 410 (the remainder is 0, so 3 and 410 are divisors of 1230)
...
1230 / 34 = 36.176470588235 (the remainder is 6, so 34 is not a divisor of 1230)
1230 / 35 = 35.142857142857 (the remainder is 5, so 35 is not a divisor of 1230)