What are the divisors of 699?

1, 3, 233, 699

There is a total of 4 positive divisors.
The sum of these divisors is 936.
The arithmetic mean is 234.

4 odd divisors

1, 3, 233, 699

How to compute the divisors of 699?

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 699 by each of the numbers from 1 to 699 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)

699 / 1 = 699 (the remainder is 0, so 1 is a divisor of 699)
699 / 2 = 349.5 (the remainder is 1, so 2 is not a divisor of 699)
699 / 3 = 233 (the remainder is 0, so 3 is a divisor of 699)
...
699 / 698 = 1.0014326647564 (the remainder is 1, so 698 is not a divisor of 699)
699 / 699 = 1 (the remainder is 0, so 699 is a divisor of 699)

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 699 (i.e. 26.438608132805). 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 $\frac{N}{d} = D$ )

699 / 1 = 699 (the remainder is 0, so 1 and 699 are divisors of 699)
699 / 2 = 349.5 (the remainder is 1, so 2 is not a divisor of 699)
699 / 3 = 233 (the remainder is 0, so 3 and 233 are divisors of 699)
...
699 / 25 = 27.96 (the remainder is 24, so 25 is not a divisor of 699)
699 / 26 = 26.884615384615 (the remainder is 23, so 26 is not a divisor of 699)