What are the divisors of 3678?

1, 2, 3, 6, 613, 1226, 1839, 3678

There is a total of 8 positive divisors.
The sum of these divisors is 7368.
The arithmetic mean is 921.

4 even divisors

2, 6, 1226, 3678

4 odd divisors

1, 3, 613, 1839

How to compute the divisors of 3678?

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 3678 by each of the numbers from 1 to 3678 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)

3678 / 1 = 3678 (the remainder is 0, so 1 is a divisor of 3678)
3678 / 2 = 1839 (the remainder is 0, so 2 is a divisor of 3678)
3678 / 3 = 1226 (the remainder is 0, so 3 is a divisor of 3678)
...
3678 / 3677 = 1.0002719608376 (the remainder is 1, so 3677 is not a divisor of 3678)
3678 / 3678 = 1 (the remainder is 0, so 3678 is a divisor of 3678)

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 3678 (i.e. 60.646516800225). 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 $\frac{N}{d} = D$ )

3678 / 1 = 3678 (the remainder is 0, so 1 and 3678 are divisors of 3678)
3678 / 2 = 1839 (the remainder is 0, so 2 and 1839 are divisors of 3678)
3678 / 3 = 1226 (the remainder is 0, so 3 and 1226 are divisors of 3678)
...
3678 / 59 = 62.338983050847 (the remainder is 20, so 59 is not a divisor of 3678)
3678 / 60 = 61.3 (the remainder is 18, so 60 is not a divisor of 3678)