What are the divisors of 1647?

1, 3, 9, 27, 61, 183, 549, 1647

There is a total of 8 positive divisors.
The sum of these divisors is 2480.
The arithmetic mean is 310.

8 odd divisors

1, 3, 9, 27, 61, 183, 549, 1647

How to compute the divisors of 1647?

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 1647 by each of the numbers from 1 to 1647 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)

1647 / 1 = 1647 (the remainder is 0, so 1 is a divisor of 1647)
1647 / 2 = 823.5 (the remainder is 1, so 2 is not a divisor of 1647)
1647 / 3 = 549 (the remainder is 0, so 3 is a divisor of 1647)
...
1647 / 1646 = 1.0006075334143 (the remainder is 1, so 1646 is not a divisor of 1647)
1647 / 1647 = 1 (the remainder is 0, so 1647 is a divisor of 1647)

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 1647 (i.e. 40.583247775406). 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 $\frac{N}{d} = D$ )

1647 / 1 = 1647 (the remainder is 0, so 1 and 1647 are divisors of 1647)
1647 / 2 = 823.5 (the remainder is 1, so 2 is not a divisor of 1647)
1647 / 3 = 549 (the remainder is 0, so 3 and 549 are divisors of 1647)
...
1647 / 39 = 42.230769230769 (the remainder is 9, so 39 is not a divisor of 1647)
1647 / 40 = 41.175 (the remainder is 7, so 40 is not a divisor of 1647)