What are the divisors of 1695?

1, 3, 5, 15, 113, 339, 565, 1695

There is a total of 8 positive divisors.
The sum of these divisors is 2736.
The arithmetic mean is 342.

8 odd divisors

1, 3, 5, 15, 113, 339, 565, 1695

How to compute the divisors of 1695?

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 1695 by each of the numbers from 1 to 1695 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)

1695 / 1 = 1695 (the remainder is 0, so 1 is a divisor of 1695)
1695 / 2 = 847.5 (the remainder is 1, so 2 is not a divisor of 1695)
1695 / 3 = 565 (the remainder is 0, so 3 is a divisor of 1695)
...
1695 / 1694 = 1.0005903187721 (the remainder is 1, so 1694 is not a divisor of 1695)
1695 / 1695 = 1 (the remainder is 0, so 1695 is a divisor of 1695)

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 1695 (i.e. 41.170377700478). 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 $\frac{N}{d} = D$ )

1695 / 1 = 1695 (the remainder is 0, so 1 and 1695 are divisors of 1695)
1695 / 2 = 847.5 (the remainder is 1, so 2 is not a divisor of 1695)
1695 / 3 = 565 (the remainder is 0, so 3 and 565 are divisors of 1695)
...
1695 / 40 = 42.375 (the remainder is 15, so 40 is not a divisor of 1695)
1695 / 41 = 41.341463414634 (the remainder is 14, so 41 is not a divisor of 1695)