What are the divisors of 1814?

1, 2, 907, 1814

There is a total of 4 positive divisors.
The sum of these divisors is 2724.
The arithmetic mean is 681.

2 even divisors

2, 1814

2 odd divisors

1, 907

How to compute the divisors of 1814?

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 1814 by each of the numbers from 1 to 1814 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)

1814 / 1 = 1814 (the remainder is 0, so 1 is a divisor of 1814)
1814 / 2 = 907 (the remainder is 0, so 2 is a divisor of 1814)
1814 / 3 = 604.66666666667 (the remainder is 2, so 3 is not a divisor of 1814)
...
1814 / 1813 = 1.0005515719801 (the remainder is 1, so 1813 is not a divisor of 1814)
1814 / 1814 = 1 (the remainder is 0, so 1814 is a divisor of 1814)

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 1814 (i.e. 42.591078878094). 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 $\frac{N}{d} = D$ )

1814 / 1 = 1814 (the remainder is 0, so 1 and 1814 are divisors of 1814)
1814 / 2 = 907 (the remainder is 0, so 2 and 907 are divisors of 1814)
1814 / 3 = 604.66666666667 (the remainder is 2, so 3 is not a divisor of 1814)
...
1814 / 41 = 44.243902439024 (the remainder is 10, so 41 is not a divisor of 1814)
1814 / 42 = 43.190476190476 (the remainder is 8, so 42 is not a divisor of 1814)