What are the divisors of 668?

1, 2, 4, 167, 334, 668

There is a total of 6 positive divisors.
The sum of these divisors is 1176.
The arithmetic mean is 196.

4 even divisors

2, 4, 334, 668

2 odd divisors

1, 167

How to compute the divisors of 668?

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 668 by each of the numbers from 1 to 668 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)

668 / 1 = 668 (the remainder is 0, so 1 is a divisor of 668)
668 / 2 = 334 (the remainder is 0, so 2 is a divisor of 668)
668 / 3 = 222.66666666667 (the remainder is 2, so 3 is not a divisor of 668)
...
668 / 667 = 1.0014992503748 (the remainder is 1, so 667 is not a divisor of 668)
668 / 668 = 1 (the remainder is 0, so 668 is a divisor of 668)

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 668 (i.e. 25.84569596664). 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 $\frac{N}{d} = D$ )

668 / 1 = 668 (the remainder is 0, so 1 and 668 are divisors of 668)
668 / 2 = 334 (the remainder is 0, so 2 and 334 are divisors of 668)
668 / 3 = 222.66666666667 (the remainder is 2, so 3 is not a divisor of 668)
...
668 / 24 = 27.833333333333 (the remainder is 20, so 24 is not a divisor of 668)
668 / 25 = 26.72 (the remainder is 18, so 25 is not a divisor of 668)