What are the divisors of 645?

1, 3, 5, 15, 43, 129, 215, 645

There is a total of 8 positive divisors.
The sum of these divisors is 1056.
The arithmetic mean is 132.

8 odd divisors

1, 3, 5, 15, 43, 129, 215, 645

How to compute the divisors of 645?

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 645 by each of the numbers from 1 to 645 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)

645 / 1 = 645 (the remainder is 0, so 1 is a divisor of 645)
645 / 2 = 322.5 (the remainder is 1, so 2 is not a divisor of 645)
645 / 3 = 215 (the remainder is 0, so 3 is a divisor of 645)
...
645 / 644 = 1.0015527950311 (the remainder is 1, so 644 is not a divisor of 645)
645 / 645 = 1 (the remainder is 0, so 645 is a divisor of 645)

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 645 (i.e. 25.396850198401). 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 $\frac{N}{d} = D$ )

645 / 1 = 645 (the remainder is 0, so 1 and 645 are divisors of 645)
645 / 2 = 322.5 (the remainder is 1, so 2 is not a divisor of 645)
645 / 3 = 215 (the remainder is 0, so 3 and 215 are divisors of 645)
...
645 / 24 = 26.875 (the remainder is 21, so 24 is not a divisor of 645)
645 / 25 = 25.8 (the remainder is 20, so 25 is not a divisor of 645)