What are the divisors of 207?

1, 3, 9, 23, 69, 207

There is a total of 6 positive divisors.
The sum of these divisors is 312.
The arithmetic mean is 52.

6 odd divisors

1, 3, 9, 23, 69, 207

How to compute the divisors of 207?

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 207 by each of the numbers from 1 to 207 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)

207 / 1 = 207 (the remainder is 0, so 1 is a divisor of 207)
207 / 2 = 103.5 (the remainder is 1, so 2 is not a divisor of 207)
207 / 3 = 69 (the remainder is 0, so 3 is a divisor of 207)
...
207 / 206 = 1.004854368932 (the remainder is 1, so 206 is not a divisor of 207)
207 / 207 = 1 (the remainder is 0, so 207 is a divisor of 207)

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 207 (i.e. 14.387494569938). 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 $\frac{N}{d} = D$ )

207 / 1 = 207 (the remainder is 0, so 1 and 207 are divisors of 207)
207 / 2 = 103.5 (the remainder is 1, so 2 is not a divisor of 207)
207 / 3 = 69 (the remainder is 0, so 3 and 69 are divisors of 207)
...
207 / 13 = 15.923076923077 (the remainder is 12, so 13 is not a divisor of 207)
207 / 14 = 14.785714285714 (the remainder is 11, so 14 is not a divisor of 207)