What are the divisors of 3693?

1, 3, 1231, 3693

There is a total of 4 positive divisors.
The sum of these divisors is 4928.
The arithmetic mean is 1232.

4 odd divisors

1, 3, 1231, 3693

How to compute the divisors of 3693?

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 3693 by each of the numbers from 1 to 3693 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)

3693 / 1 = 3693 (the remainder is 0, so 1 is a divisor of 3693)
3693 / 2 = 1846.5 (the remainder is 1, so 2 is not a divisor of 3693)
3693 / 3 = 1231 (the remainder is 0, so 3 is a divisor of 3693)
...
3693 / 3692 = 1.0002708559047 (the remainder is 1, so 3692 is not a divisor of 3693)
3693 / 3693 = 1 (the remainder is 0, so 3693 is a divisor of 3693)

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 3693 (i.e. 60.770058416954). 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 $\frac{N}{d} = D$ )

3693 / 1 = 3693 (the remainder is 0, so 1 and 3693 are divisors of 3693)
3693 / 2 = 1846.5 (the remainder is 1, so 2 is not a divisor of 3693)
3693 / 3 = 1231 (the remainder is 0, so 3 and 1231 are divisors of 3693)
...
3693 / 59 = 62.593220338983 (the remainder is 35, so 59 is not a divisor of 3693)
3693 / 60 = 61.55 (the remainder is 33, so 60 is not a divisor of 3693)