What are the divisors of 3681?

1, 3, 9, 409, 1227, 3681

There is a total of 6 positive divisors.
The sum of these divisors is 5330.
The arithmetic mean is 888.33333333333.

6 odd divisors

1, 3, 9, 409, 1227, 3681

How to compute the divisors of 3681?

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 3681 by each of the numbers from 1 to 3681 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)

3681 / 1 = 3681 (the remainder is 0, so 1 is a divisor of 3681)
3681 / 2 = 1840.5 (the remainder is 1, so 2 is not a divisor of 3681)
3681 / 3 = 1227 (the remainder is 0, so 3 is a divisor of 3681)
...
3681 / 3680 = 1.0002717391304 (the remainder is 1, so 3680 is not a divisor of 3681)
3681 / 3681 = 1 (the remainder is 0, so 3681 is a divisor of 3681)

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 3681 (i.e. 60.67124524847). 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 $\frac{N}{d} = D$ )

3681 / 1 = 3681 (the remainder is 0, so 1 and 3681 are divisors of 3681)
3681 / 2 = 1840.5 (the remainder is 1, so 2 is not a divisor of 3681)
3681 / 3 = 1227 (the remainder is 0, so 3 and 1227 are divisors of 3681)
...
3681 / 59 = 62.389830508475 (the remainder is 23, so 59 is not a divisor of 3681)
3681 / 60 = 61.35 (the remainder is 21, so 60 is not a divisor of 3681)