What are the divisors of 1648?

1, 2, 4, 8, 16, 103, 206, 412, 824, 1648

There is a total of 10 positive divisors.
The sum of these divisors is 3224.
The arithmetic mean is 322.4.

8 even divisors

2, 4, 8, 16, 206, 412, 824, 1648

2 odd divisors

1, 103

How to compute the divisors of 1648?

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 1648 by each of the numbers from 1 to 1648 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)

1648 / 1 = 1648 (the remainder is 0, so 1 is a divisor of 1648)
1648 / 2 = 824 (the remainder is 0, so 2 is a divisor of 1648)
1648 / 3 = 549.33333333333 (the remainder is 1, so 3 is not a divisor of 1648)
...
1648 / 1647 = 1.0006071645416 (the remainder is 1, so 1647 is not a divisor of 1648)
1648 / 1648 = 1 (the remainder is 0, so 1648 is a divisor of 1648)

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 1648 (i.e. 40.595566260369). 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 $\frac{N}{d} = D$ )

1648 / 1 = 1648 (the remainder is 0, so 1 and 1648 are divisors of 1648)
1648 / 2 = 824 (the remainder is 0, so 2 and 824 are divisors of 1648)
1648 / 3 = 549.33333333333 (the remainder is 1, so 3 is not a divisor of 1648)
...
1648 / 39 = 42.25641025641 (the remainder is 10, so 39 is not a divisor of 1648)
1648 / 40 = 41.2 (the remainder is 8, so 40 is not a divisor of 1648)