What are the divisors of 621?

1, 3, 9, 23, 27, 69, 207, 621

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, 9, 23, 27, 69, 207, 621

How to compute the divisors of 621?

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 621 by each of the numbers from 1 to 621 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)

621 / 1 = 621 (the remainder is 0, so 1 is a divisor of 621)
621 / 2 = 310.5 (the remainder is 1, so 2 is not a divisor of 621)
621 / 3 = 207 (the remainder is 0, so 3 is a divisor of 621)
...
621 / 620 = 1.0016129032258 (the remainder is 1, so 620 is not a divisor of 621)
621 / 621 = 1 (the remainder is 0, so 621 is a divisor of 621)

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 621 (i.e. 24.919871588754). 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 $\frac{N}{d} = D$ )

621 / 1 = 621 (the remainder is 0, so 1 and 621 are divisors of 621)
621 / 2 = 310.5 (the remainder is 1, so 2 is not a divisor of 621)
621 / 3 = 207 (the remainder is 0, so 3 and 207 are divisors of 621)
...
621 / 23 = 27 (the remainder is 0, so 23 and 27 are divisors of 621)
621 / 24 = 25.875 (the remainder is 21, so 24 is not a divisor of 621)