What are the divisors of 209?

1, 11, 19, 209

There is a total of 4 positive divisors.
The sum of these divisors is 240.
The arithmetic mean is 60.

4 odd divisors

1, 11, 19, 209

How to compute the divisors of 209?

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 209 by each of the numbers from 1 to 209 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)

209 / 1 = 209 (the remainder is 0, so 1 is a divisor of 209)
209 / 2 = 104.5 (the remainder is 1, so 2 is not a divisor of 209)
209 / 3 = 69.666666666667 (the remainder is 2, so 3 is not a divisor of 209)
...
209 / 208 = 1.0048076923077 (the remainder is 1, so 208 is not a divisor of 209)
209 / 209 = 1 (the remainder is 0, so 209 is a divisor of 209)

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 209 (i.e. 14.456832294801). 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 $\frac{N}{d} = D$ )

209 / 1 = 209 (the remainder is 0, so 1 and 209 are divisors of 209)
209 / 2 = 104.5 (the remainder is 1, so 2 is not a divisor of 209)
209 / 3 = 69.666666666667 (the remainder is 2, so 3 is not a divisor of 209)
...
209 / 13 = 16.076923076923 (the remainder is 1, so 13 is not a divisor of 209)
209 / 14 = 14.928571428571 (the remainder is 13, so 14 is not a divisor of 209)