What are the divisors of 615?

1, 3, 5, 15, 41, 123, 205, 615

There is a total of 8 positive divisors.
The sum of these divisors is 1008.
The arithmetic mean is 126.

8 odd divisors

1, 3, 5, 15, 41, 123, 205, 615

How to compute the divisors of 615?

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 615 by each of the numbers from 1 to 615 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)

615 / 1 = 615 (the remainder is 0, so 1 is a divisor of 615)
615 / 2 = 307.5 (the remainder is 1, so 2 is not a divisor of 615)
615 / 3 = 205 (the remainder is 0, so 3 is a divisor of 615)
...
615 / 614 = 1.0016286644951 (the remainder is 1, so 614 is not a divisor of 615)
615 / 615 = 1 (the remainder is 0, so 615 is a divisor of 615)

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 615 (i.e. 24.799193535274). 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 $\frac{N}{d} = D$ )

615 / 1 = 615 (the remainder is 0, so 1 and 615 are divisors of 615)
615 / 2 = 307.5 (the remainder is 1, so 2 is not a divisor of 615)
615 / 3 = 205 (the remainder is 0, so 3 and 205 are divisors of 615)
...
615 / 23 = 26.739130434783 (the remainder is 17, so 23 is not a divisor of 615)
615 / 24 = 25.625 (the remainder is 15, so 24 is not a divisor of 615)