What are the divisors of 1723?

1, 1723

There is a total of 2 positive divisors.
The sum of these divisors is 1724.
The arithmetic mean is 862.

2 odd divisors

1, 1723

How to compute the divisors of 1723?

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 1723 by each of the numbers from 1 to 1723 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)

1723 / 1 = 1723 (the remainder is 0, so 1 is a divisor of 1723)
1723 / 2 = 861.5 (the remainder is 1, so 2 is not a divisor of 1723)
1723 / 3 = 574.33333333333 (the remainder is 1, so 3 is not a divisor of 1723)
...
1723 / 1722 = 1.0005807200929 (the remainder is 1, so 1722 is not a divisor of 1723)
1723 / 1723 = 1 (the remainder is 0, so 1723 is a divisor of 1723)

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 1723 (i.e. 41.509035161035). 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 $\frac{N}{d} = D$ )

1723 / 1 = 1723 (the remainder is 0, so 1 and 1723 are divisors of 1723)
1723 / 2 = 861.5 (the remainder is 1, so 2 is not a divisor of 1723)
1723 / 3 = 574.33333333333 (the remainder is 1, so 3 is not a divisor of 1723)
...
1723 / 40 = 43.075 (the remainder is 3, so 40 is not a divisor of 1723)
1723 / 41 = 42.024390243902 (the remainder is 1, so 41 is not a divisor of 1723)