What are the divisors of 609?

1, 3, 7, 21, 29, 87, 203, 609

There is a total of 8 positive divisors.
The sum of these divisors is 960.
The arithmetic mean is 120.

8 odd divisors

1, 3, 7, 21, 29, 87, 203, 609

How to compute the divisors of 609?

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 609 by each of the numbers from 1 to 609 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)

609 / 1 = 609 (the remainder is 0, so 1 is a divisor of 609)
609 / 2 = 304.5 (the remainder is 1, so 2 is not a divisor of 609)
609 / 3 = 203 (the remainder is 0, so 3 is a divisor of 609)
...
609 / 608 = 1.0016447368421 (the remainder is 1, so 608 is not a divisor of 609)
609 / 609 = 1 (the remainder is 0, so 609 is a divisor of 609)

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 609 (i.e. 24.677925358506). 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 $\frac{N}{d} = D$ )

609 / 1 = 609 (the remainder is 0, so 1 and 609 are divisors of 609)
609 / 2 = 304.5 (the remainder is 1, so 2 is not a divisor of 609)
609 / 3 = 203 (the remainder is 0, so 3 and 203 are divisors of 609)
...
609 / 23 = 26.478260869565 (the remainder is 11, so 23 is not a divisor of 609)
609 / 24 = 25.375 (the remainder is 9, so 24 is not a divisor of 609)