What are the divisors of 3653?

1, 13, 281, 3653

There is a total of 4 positive divisors.
The sum of these divisors is 3948.
The arithmetic mean is 987.

4 odd divisors

1, 13, 281, 3653

How to compute the divisors of 3653?

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 3653 by each of the numbers from 1 to 3653 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)

3653 / 1 = 3653 (the remainder is 0, so 1 is a divisor of 3653)
3653 / 2 = 1826.5 (the remainder is 1, so 2 is not a divisor of 3653)
3653 / 3 = 1217.6666666667 (the remainder is 2, so 3 is not a divisor of 3653)
...
3653 / 3652 = 1.000273822563 (the remainder is 1, so 3652 is not a divisor of 3653)
3653 / 3653 = 1 (the remainder is 0, so 3653 is a divisor of 3653)

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 3653 (i.e. 60.440052945046). 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 $\frac{N}{d} = D$ )

3653 / 1 = 3653 (the remainder is 0, so 1 and 3653 are divisors of 3653)
3653 / 2 = 1826.5 (the remainder is 1, so 2 is not a divisor of 3653)
3653 / 3 = 1217.6666666667 (the remainder is 2, so 3 is not a divisor of 3653)
...
3653 / 59 = 61.915254237288 (the remainder is 54, so 59 is not a divisor of 3653)
3653 / 60 = 60.883333333333 (the remainder is 53, so 60 is not a divisor of 3653)