What are the divisors of 3697?

1, 3697

There is a total of 2 positive divisors.
The sum of these divisors is 3698.
The arithmetic mean is 1849.

2 odd divisors

1, 3697

How to compute the divisors of 3697?

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 3697 by each of the numbers from 1 to 3697 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)

3697 / 1 = 3697 (the remainder is 0, so 1 is a divisor of 3697)
3697 / 2 = 1848.5 (the remainder is 1, so 2 is not a divisor of 3697)
3697 / 3 = 1232.3333333333 (the remainder is 1, so 3 is not a divisor of 3697)
...
3697 / 3696 = 1.0002705627706 (the remainder is 1, so 3696 is not a divisor of 3697)
3697 / 3697 = 1 (the remainder is 0, so 3697 is a divisor of 3697)

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 3697 (i.e. 60.802960454241). 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 $\frac{N}{d} = D$ )

3697 / 1 = 3697 (the remainder is 0, so 1 and 3697 are divisors of 3697)
3697 / 2 = 1848.5 (the remainder is 1, so 2 is not a divisor of 3697)
3697 / 3 = 1232.3333333333 (the remainder is 1, so 3 is not a divisor of 3697)
...
3697 / 59 = 62.661016949153 (the remainder is 39, so 59 is not a divisor of 3697)
3697 / 60 = 61.616666666667 (the remainder is 37, so 60 is not a divisor of 3697)