What are the divisors of 3727?

1, 3727

There is a total of 2 positive divisors.
The sum of these divisors is 3728.
The arithmetic mean is 1864.

2 odd divisors

1, 3727

How to compute the divisors of 3727?

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 3727 by each of the numbers from 1 to 3727 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)

3727 / 1 = 3727 (the remainder is 0, so 1 is a divisor of 3727)
3727 / 2 = 1863.5 (the remainder is 1, so 2 is not a divisor of 3727)
3727 / 3 = 1242.3333333333 (the remainder is 1, so 3 is not a divisor of 3727)
...
3727 / 3726 = 1.0002683843264 (the remainder is 1, so 3726 is not a divisor of 3727)
3727 / 3727 = 1 (the remainder is 0, so 3727 is a divisor of 3727)

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 3727 (i.e. 61.049160518389). 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 $\frac{N}{d} = D$ )

3727 / 1 = 3727 (the remainder is 0, so 1 and 3727 are divisors of 3727)
3727 / 2 = 1863.5 (the remainder is 1, so 2 is not a divisor of 3727)
3727 / 3 = 1242.3333333333 (the remainder is 1, so 3 is not a divisor of 3727)
...
3727 / 60 = 62.116666666667 (the remainder is 7, so 60 is not a divisor of 3727)
3727 / 61 = 61.098360655738 (the remainder is 6, so 61 is not a divisor of 3727)