What are the divisors of 678?

1, 2, 3, 6, 113, 226, 339, 678

There is a total of 8 positive divisors.
The sum of these divisors is 1368.
The arithmetic mean is 171.

4 even divisors

2, 6, 226, 678

4 odd divisors

1, 3, 113, 339

How to compute the divisors of 678?

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 678 by each of the numbers from 1 to 678 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)

678 / 1 = 678 (the remainder is 0, so 1 is a divisor of 678)
678 / 2 = 339 (the remainder is 0, so 2 is a divisor of 678)
678 / 3 = 226 (the remainder is 0, so 3 is a divisor of 678)
...
678 / 677 = 1.0014771048744 (the remainder is 1, so 677 is not a divisor of 678)
678 / 678 = 1 (the remainder is 0, so 678 is a divisor of 678)

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 678 (i.e. 26.038433132583). 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 $\frac{N}{d} = D$ )

678 / 1 = 678 (the remainder is 0, so 1 and 678 are divisors of 678)
678 / 2 = 339 (the remainder is 0, so 2 and 339 are divisors of 678)
678 / 3 = 226 (the remainder is 0, so 3 and 226 are divisors of 678)
...
678 / 25 = 27.12 (the remainder is 3, so 25 is not a divisor of 678)
678 / 26 = 26.076923076923 (the remainder is 2, so 26 is not a divisor of 678)