What are the divisors of 1693?

1, 1693

There is a total of 2 positive divisors.
The sum of these divisors is 1694.
The arithmetic mean is 847.

2 odd divisors

1, 1693

How to compute the divisors of 1693?

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 1693 by each of the numbers from 1 to 1693 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)

1693 / 1 = 1693 (the remainder is 0, so 1 is a divisor of 1693)
1693 / 2 = 846.5 (the remainder is 1, so 2 is not a divisor of 1693)
1693 / 3 = 564.33333333333 (the remainder is 1, so 3 is not a divisor of 1693)
...
1693 / 1692 = 1.0005910165485 (the remainder is 1, so 1692 is not a divisor of 1693)
1693 / 1693 = 1 (the remainder is 0, so 1693 is a divisor of 1693)

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 1693 (i.e. 41.146081222882). 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 $\frac{N}{d} = D$ )

1693 / 1 = 1693 (the remainder is 0, so 1 and 1693 are divisors of 1693)
1693 / 2 = 846.5 (the remainder is 1, so 2 is not a divisor of 1693)
1693 / 3 = 564.33333333333 (the remainder is 1, so 3 is not a divisor of 1693)
...
1693 / 40 = 42.325 (the remainder is 13, so 40 is not a divisor of 1693)
1693 / 41 = 41.292682926829 (the remainder is 12, so 41 is not a divisor of 1693)