What are the divisors of 1726?

1, 2, 863, 1726

There is a total of 4 positive divisors.
The sum of these divisors is 2592.
The arithmetic mean is 648.

2 even divisors

2, 1726

2 odd divisors

1, 863

How to compute the divisors of 1726?

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 1726 by each of the numbers from 1 to 1726 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)

1726 / 1 = 1726 (the remainder is 0, so 1 is a divisor of 1726)
1726 / 2 = 863 (the remainder is 0, so 2 is a divisor of 1726)
1726 / 3 = 575.33333333333 (the remainder is 1, so 3 is not a divisor of 1726)
...
1726 / 1725 = 1.0005797101449 (the remainder is 1, so 1725 is not a divisor of 1726)
1726 / 1726 = 1 (the remainder is 0, so 1726 is a divisor of 1726)

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 1726 (i.e. 41.545156155682). 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 $\frac{N}{d} = D$ )

1726 / 1 = 1726 (the remainder is 0, so 1 and 1726 are divisors of 1726)
1726 / 2 = 863 (the remainder is 0, so 2 and 863 are divisors of 1726)
1726 / 3 = 575.33333333333 (the remainder is 1, so 3 is not a divisor of 1726)
...
1726 / 40 = 43.15 (the remainder is 6, so 40 is not a divisor of 1726)
1726 / 41 = 42.09756097561 (the remainder is 4, so 41 is not a divisor of 1726)