What are the divisors of 315?

1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315

There is a total of 12 positive divisors.
The sum of these divisors is 624.
The arithmetic mean is 52.

12 odd divisors

1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315

How to compute the divisors of 315?

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 315 by each of the numbers from 1 to 315 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)

315 / 1 = 315 (the remainder is 0, so 1 is a divisor of 315)
315 / 2 = 157.5 (the remainder is 1, so 2 is not a divisor of 315)
315 / 3 = 105 (the remainder is 0, so 3 is a divisor of 315)
...
315 / 314 = 1.0031847133758 (the remainder is 1, so 314 is not a divisor of 315)
315 / 315 = 1 (the remainder is 0, so 315 is a divisor of 315)

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 315 (i.e. 17.748239349299). 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 $\frac{N}{d} = D$ )

315 / 1 = 315 (the remainder is 0, so 1 and 315 are divisors of 315)
315 / 2 = 157.5 (the remainder is 1, so 2 is not a divisor of 315)
315 / 3 = 105 (the remainder is 0, so 3 and 105 are divisors of 315)
...
315 / 16 = 19.6875 (the remainder is 11, so 16 is not a divisor of 315)
315 / 17 = 18.529411764706 (the remainder is 9, so 17 is not a divisor of 315)