What are the divisors of 3731?

1, 7, 13, 41, 91, 287, 533, 3731

There is a total of 8 positive divisors.
The sum of these divisors is 4704.
The arithmetic mean is 588.

8 odd divisors

1, 7, 13, 41, 91, 287, 533, 3731

How to compute the divisors of 3731?

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 3731 by each of the numbers from 1 to 3731 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)

3731 / 1 = 3731 (the remainder is 0, so 1 is a divisor of 3731)
3731 / 2 = 1865.5 (the remainder is 1, so 2 is not a divisor of 3731)
3731 / 3 = 1243.6666666667 (the remainder is 2, so 3 is not a divisor of 3731)
...
3731 / 3730 = 1.0002680965147 (the remainder is 1, so 3730 is not a divisor of 3731)
3731 / 3731 = 1 (the remainder is 0, so 3731 is a divisor of 3731)

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 3731 (i.e. 61.081912216302). 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 $\frac{N}{d} = D$ )

3731 / 1 = 3731 (the remainder is 0, so 1 and 3731 are divisors of 3731)
3731 / 2 = 1865.5 (the remainder is 1, so 2 is not a divisor of 3731)
3731 / 3 = 1243.6666666667 (the remainder is 2, so 3 is not a divisor of 3731)
...
3731 / 60 = 62.183333333333 (the remainder is 11, so 60 is not a divisor of 3731)
3731 / 61 = 61.16393442623 (the remainder is 10, so 61 is not a divisor of 3731)