What are the divisors of 830?

1, 2, 5, 10, 83, 166, 415, 830

There is a total of 8 positive divisors.
The sum of these divisors is 1512.
The arithmetic mean is 189.

4 even divisors

2, 10, 166, 830

4 odd divisors

1, 5, 83, 415

How to compute the divisors of 830?

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 830 by each of the numbers from 1 to 830 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)

830 / 1 = 830 (the remainder is 0, so 1 is a divisor of 830)
830 / 2 = 415 (the remainder is 0, so 2 is a divisor of 830)
830 / 3 = 276.66666666667 (the remainder is 2, so 3 is not a divisor of 830)
...
830 / 829 = 1.0012062726176 (the remainder is 1, so 829 is not a divisor of 830)
830 / 830 = 1 (the remainder is 0, so 830 is a divisor of 830)

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 830 (i.e. 28.809720581776). 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 $\frac{N}{d} = D$ )

830 / 1 = 830 (the remainder is 0, so 1 and 830 are divisors of 830)
830 / 2 = 415 (the remainder is 0, so 2 and 415 are divisors of 830)
830 / 3 = 276.66666666667 (the remainder is 2, so 3 is not a divisor of 830)
...
830 / 27 = 30.740740740741 (the remainder is 20, so 27 is not a divisor of 830)
830 / 28 = 29.642857142857 (the remainder is 18, so 28 is not a divisor of 830)