What are the divisors of 646?

1, 2, 17, 19, 34, 38, 323, 646

There is a total of 8 positive divisors.
The sum of these divisors is 1080.
The arithmetic mean is 135.

4 even divisors

2, 34, 38, 646

4 odd divisors

1, 17, 19, 323

How to compute the divisors of 646?

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 646 by each of the numbers from 1 to 646 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)

646 / 1 = 646 (the remainder is 0, so 1 is a divisor of 646)
646 / 2 = 323 (the remainder is 0, so 2 is a divisor of 646)
646 / 3 = 215.33333333333 (the remainder is 1, so 3 is not a divisor of 646)
...
646 / 645 = 1.0015503875969 (the remainder is 1, so 645 is not a divisor of 646)
646 / 646 = 1 (the remainder is 0, so 646 is a divisor of 646)

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 646 (i.e. 25.416530054278). 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 $\frac{N}{d} = D$ )

646 / 1 = 646 (the remainder is 0, so 1 and 646 are divisors of 646)
646 / 2 = 323 (the remainder is 0, so 2 and 323 are divisors of 646)
646 / 3 = 215.33333333333 (the remainder is 1, so 3 is not a divisor of 646)
...
646 / 24 = 26.916666666667 (the remainder is 22, so 24 is not a divisor of 646)
646 / 25 = 25.84 (the remainder is 21, so 25 is not a divisor of 646)