What are the divisors of 679?

1, 7, 97, 679

There is a total of 4 positive divisors.
The sum of these divisors is 784.
The arithmetic mean is 196.

4 odd divisors

1, 7, 97, 679

How to compute the divisors of 679?

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 679 by each of the numbers from 1 to 679 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)

679 / 1 = 679 (the remainder is 0, so 1 is a divisor of 679)
679 / 2 = 339.5 (the remainder is 1, so 2 is not a divisor of 679)
679 / 3 = 226.33333333333 (the remainder is 1, so 3 is not a divisor of 679)
...
679 / 678 = 1.0014749262537 (the remainder is 1, so 678 is not a divisor of 679)
679 / 679 = 1 (the remainder is 0, so 679 is a divisor of 679)

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 679 (i.e. 26.057628441591). 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 $\frac{N}{d} = D$ )

679 / 1 = 679 (the remainder is 0, so 1 and 679 are divisors of 679)
679 / 2 = 339.5 (the remainder is 1, so 2 is not a divisor of 679)
679 / 3 = 226.33333333333 (the remainder is 1, so 3 is not a divisor of 679)
...
679 / 25 = 27.16 (the remainder is 4, so 25 is not a divisor of 679)
679 / 26 = 26.115384615385 (the remainder is 3, so 26 is not a divisor of 679)