What are the divisors of 195?

1, 3, 5, 13, 15, 39, 65, 195

There is a total of 8 positive divisors.
The sum of these divisors is 336.
The arithmetic mean is 42.

8 odd divisors

1, 3, 5, 13, 15, 39, 65, 195

How to compute the divisors of 195?

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 195 by each of the numbers from 1 to 195 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)

195 / 1 = 195 (the remainder is 0, so 1 is a divisor of 195)
195 / 2 = 97.5 (the remainder is 1, so 2 is not a divisor of 195)
195 / 3 = 65 (the remainder is 0, so 3 is a divisor of 195)
...
195 / 194 = 1.0051546391753 (the remainder is 1, so 194 is not a divisor of 195)
195 / 195 = 1 (the remainder is 0, so 195 is a divisor of 195)

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 195 (i.e. 13.964240043769). 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 $\frac{N}{d} = D$ )

195 / 1 = 195 (the remainder is 0, so 1 and 195 are divisors of 195)
195 / 2 = 97.5 (the remainder is 1, so 2 is not a divisor of 195)
195 / 3 = 65 (the remainder is 0, so 3 and 65 are divisors of 195)
...
195 / 12 = 16.25 (the remainder is 3, so 12 is not a divisor of 195)
195 / 13 = 15 (the remainder is 0, so 13 and 15 are divisors of 195)