What are the divisors of 1731?

1, 3, 577, 1731

There is a total of 4 positive divisors.
The sum of these divisors is 2312.
The arithmetic mean is 578.

4 odd divisors

1, 3, 577, 1731

How to compute the divisors of 1731?

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 1731 by each of the numbers from 1 to 1731 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)

1731 / 1 = 1731 (the remainder is 0, so 1 is a divisor of 1731)
1731 / 2 = 865.5 (the remainder is 1, so 2 is not a divisor of 1731)
1731 / 3 = 577 (the remainder is 0, so 3 is a divisor of 1731)
...
1731 / 1730 = 1.0005780346821 (the remainder is 1, so 1730 is not a divisor of 1731)
1731 / 1731 = 1 (the remainder is 0, so 1731 is a divisor of 1731)

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 1731 (i.e. 41.605288125429). 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 $\frac{N}{d} = D$ )

1731 / 1 = 1731 (the remainder is 0, so 1 and 1731 are divisors of 1731)
1731 / 2 = 865.5 (the remainder is 1, so 2 is not a divisor of 1731)
1731 / 3 = 577 (the remainder is 0, so 3 and 577 are divisors of 1731)
...
1731 / 40 = 43.275 (the remainder is 11, so 40 is not a divisor of 1731)
1731 / 41 = 42.219512195122 (the remainder is 9, so 41 is not a divisor of 1731)