What are the divisors of 639?

1, 3, 9, 71, 213, 639

There is a total of 6 positive divisors.
The sum of these divisors is 936.
The arithmetic mean is 156.

6 odd divisors

1, 3, 9, 71, 213, 639

How to compute the divisors of 639?

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 639 by each of the numbers from 1 to 639 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)

639 / 1 = 639 (the remainder is 0, so 1 is a divisor of 639)
639 / 2 = 319.5 (the remainder is 1, so 2 is not a divisor of 639)
639 / 3 = 213 (the remainder is 0, so 3 is a divisor of 639)
...
639 / 638 = 1.0015673981191 (the remainder is 1, so 638 is not a divisor of 639)
639 / 639 = 1 (the remainder is 0, so 639 is a divisor of 639)

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 639 (i.e. 25.278449319529). 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 $\frac{N}{d} = D$ )

639 / 1 = 639 (the remainder is 0, so 1 and 639 are divisors of 639)
639 / 2 = 319.5 (the remainder is 1, so 2 is not a divisor of 639)
639 / 3 = 213 (the remainder is 0, so 3 and 213 are divisors of 639)
...
639 / 24 = 26.625 (the remainder is 15, so 24 is not a divisor of 639)
639 / 25 = 25.56 (the remainder is 14, so 25 is not a divisor of 639)