What are the divisors of 199?

1, 199

There is a total of 2 positive divisors.
The sum of these divisors is 200.
The arithmetic mean is 100.

2 odd divisors

1, 199

How to compute the divisors of 199?

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 199 by each of the numbers from 1 to 199 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)

199 / 1 = 199 (the remainder is 0, so 1 is a divisor of 199)
199 / 2 = 99.5 (the remainder is 1, so 2 is not a divisor of 199)
199 / 3 = 66.333333333333 (the remainder is 1, so 3 is not a divisor of 199)
...
199 / 198 = 1.0050505050505 (the remainder is 1, so 198 is not a divisor of 199)
199 / 199 = 1 (the remainder is 0, so 199 is a divisor of 199)

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 199 (i.e. 14.106735979666). 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 $\frac{N}{d} = D$ )

199 / 1 = 199 (the remainder is 0, so 1 and 199 are divisors of 199)
199 / 2 = 99.5 (the remainder is 1, so 2 is not a divisor of 199)
199 / 3 = 66.333333333333 (the remainder is 1, so 3 is not a divisor of 199)
...
199 / 13 = 15.307692307692 (the remainder is 4, so 13 is not a divisor of 199)
199 / 14 = 14.214285714286 (the remainder is 3, so 14 is not a divisor of 199)