What are the divisors of 644?

1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644

There is a total of 12 positive divisors.
The sum of these divisors is 1344.
The arithmetic mean is 112.

8 even divisors

2, 4, 14, 28, 46, 92, 322, 644

4 odd divisors

1, 7, 23, 161

How to compute the divisors of 644?

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 644 by each of the numbers from 1 to 644 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)

644 / 1 = 644 (the remainder is 0, so 1 is a divisor of 644)
644 / 2 = 322 (the remainder is 0, so 2 is a divisor of 644)
644 / 3 = 214.66666666667 (the remainder is 2, so 3 is not a divisor of 644)
...
644 / 643 = 1.0015552099533 (the remainder is 1, so 643 is not a divisor of 644)
644 / 644 = 1 (the remainder is 0, so 644 is a divisor of 644)

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 644 (i.e. 25.377155080899). 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 $\frac{N}{d} = D$ )

644 / 1 = 644 (the remainder is 0, so 1 and 644 are divisors of 644)
644 / 2 = 322 (the remainder is 0, so 2 and 322 are divisors of 644)
644 / 3 = 214.66666666667 (the remainder is 2, so 3 is not a divisor of 644)
...
644 / 24 = 26.833333333333 (the remainder is 20, so 24 is not a divisor of 644)
644 / 25 = 25.76 (the remainder is 19, so 25 is not a divisor of 644)