What are the divisors of 1679?

1, 23, 73, 1679

There is a total of 4 positive divisors.
The sum of these divisors is 1776.
The arithmetic mean is 444.

4 odd divisors

1, 23, 73, 1679

How to compute the divisors of 1679?

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 1679 by each of the numbers from 1 to 1679 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)

1679 / 1 = 1679 (the remainder is 0, so 1 is a divisor of 1679)
1679 / 2 = 839.5 (the remainder is 1, so 2 is not a divisor of 1679)
1679 / 3 = 559.66666666667 (the remainder is 2, so 3 is not a divisor of 1679)
...
1679 / 1678 = 1.0005959475566 (the remainder is 1, so 1678 is not a divisor of 1679)
1679 / 1679 = 1 (the remainder is 0, so 1679 is a divisor of 1679)

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 1679 (i.e. 40.975602497096). 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 $\frac{N}{d} = D$ )

1679 / 1 = 1679 (the remainder is 0, so 1 and 1679 are divisors of 1679)
1679 / 2 = 839.5 (the remainder is 1, so 2 is not a divisor of 1679)
1679 / 3 = 559.66666666667 (the remainder is 2, so 3 is not a divisor of 1679)
...
1679 / 39 = 43.051282051282 (the remainder is 2, so 39 is not a divisor of 1679)
1679 / 40 = 41.975 (the remainder is 39, so 40 is not a divisor of 1679)