What are the divisors of 191?

1, 191

There is a total of 2 positive divisors.
The sum of these divisors is 192.
The arithmetic mean is 96.

2 odd divisors

1, 191

How to compute the divisors of 191?

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 191 by each of the numbers from 1 to 191 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)

191 / 1 = 191 (the remainder is 0, so 1 is a divisor of 191)
191 / 2 = 95.5 (the remainder is 1, so 2 is not a divisor of 191)
191 / 3 = 63.666666666667 (the remainder is 2, so 3 is not a divisor of 191)
...
191 / 190 = 1.0052631578947 (the remainder is 1, so 190 is not a divisor of 191)
191 / 191 = 1 (the remainder is 0, so 191 is a divisor of 191)

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 191 (i.e. 13.820274961085). 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 $\frac{N}{d} = D$ )

191 / 1 = 191 (the remainder is 0, so 1 and 191 are divisors of 191)
191 / 2 = 95.5 (the remainder is 1, so 2 is not a divisor of 191)
191 / 3 = 63.666666666667 (the remainder is 2, so 3 is not a divisor of 191)
...
191 / 12 = 15.916666666667 (the remainder is 11, so 12 is not a divisor of 191)
191 / 13 = 14.692307692308 (the remainder is 9, so 13 is not a divisor of 191)