What are the divisors of 655?

1, 5, 131, 655

There is a total of 4 positive divisors.
The sum of these divisors is 792.
The arithmetic mean is 198.

4 odd divisors

1, 5, 131, 655

How to compute the divisors of 655?

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 655 by each of the numbers from 1 to 655 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)

655 / 1 = 655 (the remainder is 0, so 1 is a divisor of 655)
655 / 2 = 327.5 (the remainder is 1, so 2 is not a divisor of 655)
655 / 3 = 218.33333333333 (the remainder is 1, so 3 is not a divisor of 655)
...
655 / 654 = 1.0015290519878 (the remainder is 1, so 654 is not a divisor of 655)
655 / 655 = 1 (the remainder is 0, so 655 is a divisor of 655)

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 655 (i.e. 25.592967784139). 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 $\frac{N}{d} = D$ )

655 / 1 = 655 (the remainder is 0, so 1 and 655 are divisors of 655)
655 / 2 = 327.5 (the remainder is 1, so 2 is not a divisor of 655)
655 / 3 = 218.33333333333 (the remainder is 1, so 3 is not a divisor of 655)
...
655 / 24 = 27.291666666667 (the remainder is 7, so 24 is not a divisor of 655)
655 / 25 = 26.2 (the remainder is 5, so 25 is not a divisor of 655)