What are the divisors of 1727?

1, 11, 157, 1727

There is a total of 4 positive divisors.
The sum of these divisors is 1896.
The arithmetic mean is 474.

4 odd divisors

1, 11, 157, 1727

How to compute the divisors of 1727?

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 1727 by each of the numbers from 1 to 1727 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)

1727 / 1 = 1727 (the remainder is 0, so 1 is a divisor of 1727)
1727 / 2 = 863.5 (the remainder is 1, so 2 is not a divisor of 1727)
1727 / 3 = 575.66666666667 (the remainder is 2, so 3 is not a divisor of 1727)
...
1727 / 1726 = 1.0005793742758 (the remainder is 1, so 1726 is not a divisor of 1727)
1727 / 1727 = 1 (the remainder is 0, so 1727 is a divisor of 1727)

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 1727 (i.e. 41.55718951036). 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 $\frac{N}{d} = D$ )

1727 / 1 = 1727 (the remainder is 0, so 1 and 1727 are divisors of 1727)
1727 / 2 = 863.5 (the remainder is 1, so 2 is not a divisor of 1727)
1727 / 3 = 575.66666666667 (the remainder is 2, so 3 is not a divisor of 1727)
...
1727 / 40 = 43.175 (the remainder is 7, so 40 is not a divisor of 1727)
1727 / 41 = 42.121951219512 (the remainder is 5, so 41 is not a divisor of 1727)