What are the divisors of 1785?

1, 3, 5, 7, 15, 17, 21, 35, 51, 85, 105, 119, 255, 357, 595, 1785

There is a total of 16 positive divisors.
The sum of these divisors is 3456.
The arithmetic mean is 216.

16 odd divisors

1, 3, 5, 7, 15, 17, 21, 35, 51, 85, 105, 119, 255, 357, 595, 1785

How to compute the divisors of 1785?

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 1785 by each of the numbers from 1 to 1785 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)

1785 / 1 = 1785 (the remainder is 0, so 1 is a divisor of 1785)
1785 / 2 = 892.5 (the remainder is 1, so 2 is not a divisor of 1785)
1785 / 3 = 595 (the remainder is 0, so 3 is a divisor of 1785)
...
1785 / 1784 = 1.0005605381166 (the remainder is 1, so 1784 is not a divisor of 1785)
1785 / 1785 = 1 (the remainder is 0, so 1785 is a divisor of 1785)

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 1785 (i.e. 42.249260348555). 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 $\frac{N}{d} = D$ )

1785 / 1 = 1785 (the remainder is 0, so 1 and 1785 are divisors of 1785)
1785 / 2 = 892.5 (the remainder is 1, so 2 is not a divisor of 1785)
1785 / 3 = 595 (the remainder is 0, so 3 and 595 are divisors of 1785)
...
1785 / 41 = 43.536585365854 (the remainder is 22, so 41 is not a divisor of 1785)
1785 / 42 = 42.5 (the remainder is 21, so 42 is not a divisor of 1785)