What are the divisors of 643?

1, 643

There is a total of 2 positive divisors.
The sum of these divisors is 644.
The arithmetic mean is 322.

2 odd divisors

1, 643

How to compute the divisors of 643?

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 643 by each of the numbers from 1 to 643 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)

643 / 1 = 643 (the remainder is 0, so 1 is a divisor of 643)
643 / 2 = 321.5 (the remainder is 1, so 2 is not a divisor of 643)
643 / 3 = 214.33333333333 (the remainder is 1, so 3 is not a divisor of 643)
...
643 / 642 = 1.0015576323988 (the remainder is 1, so 642 is not a divisor of 643)
643 / 643 = 1 (the remainder is 0, so 643 is a divisor of 643)

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 643 (i.e. 25.357444666212). 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 $\frac{N}{d} = D$ )

643 / 1 = 643 (the remainder is 0, so 1 and 643 are divisors of 643)
643 / 2 = 321.5 (the remainder is 1, so 2 is not a divisor of 643)
643 / 3 = 214.33333333333 (the remainder is 1, so 3 is not a divisor of 643)
...
643 / 24 = 26.791666666667 (the remainder is 19, so 24 is not a divisor of 643)
643 / 25 = 25.72 (the remainder is 18, so 25 is not a divisor of 643)