What are the divisors of 1823?

1, 1823

There is a total of 2 positive divisors.
The sum of these divisors is 1824.
The arithmetic mean is 912.

2 odd divisors

1, 1823

How to compute the divisors of 1823?

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 1823 by each of the numbers from 1 to 1823 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)

1823 / 1 = 1823 (the remainder is 0, so 1 is a divisor of 1823)
1823 / 2 = 911.5 (the remainder is 1, so 2 is not a divisor of 1823)
1823 / 3 = 607.66666666667 (the remainder is 2, so 3 is not a divisor of 1823)
...
1823 / 1822 = 1.0005488474204 (the remainder is 1, so 1822 is not a divisor of 1823)
1823 / 1823 = 1 (the remainder is 0, so 1823 is a divisor of 1823)

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 1823 (i.e. 42.696604080418). 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 $\frac{N}{d} = D$ )

1823 / 1 = 1823 (the remainder is 0, so 1 and 1823 are divisors of 1823)
1823 / 2 = 911.5 (the remainder is 1, so 2 is not a divisor of 1823)
1823 / 3 = 607.66666666667 (the remainder is 2, so 3 is not a divisor of 1823)
...
1823 / 41 = 44.463414634146 (the remainder is 19, so 41 is not a divisor of 1823)
1823 / 42 = 43.404761904762 (the remainder is 17, so 42 is not a divisor of 1823)