What are the divisors of 667?

1, 23, 29, 667

There is a total of 4 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 180.

4 odd divisors

1, 23, 29, 667

How to compute the divisors of 667?

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 667 by each of the numbers from 1 to 667 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)

667 / 1 = 667 (the remainder is 0, so 1 is a divisor of 667)
667 / 2 = 333.5 (the remainder is 1, so 2 is not a divisor of 667)
667 / 3 = 222.33333333333 (the remainder is 1, so 3 is not a divisor of 667)
...
667 / 666 = 1.0015015015015 (the remainder is 1, so 666 is not a divisor of 667)
667 / 667 = 1 (the remainder is 0, so 667 is a divisor of 667)

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 667 (i.e. 25.82634314029). 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 $\frac{N}{d} = D$ )

667 / 1 = 667 (the remainder is 0, so 1 and 667 are divisors of 667)
667 / 2 = 333.5 (the remainder is 1, so 2 is not a divisor of 667)
667 / 3 = 222.33333333333 (the remainder is 1, so 3 is not a divisor of 667)
...
667 / 24 = 27.791666666667 (the remainder is 19, so 24 is not a divisor of 667)
667 / 25 = 26.68 (the remainder is 17, so 25 is not a divisor of 667)