What are the divisors of 190?

1, 2, 5, 10, 19, 38, 95, 190

There is a total of 8 positive divisors.
The sum of these divisors is 360.
The arithmetic mean is 45.

4 even divisors

2, 10, 38, 190

4 odd divisors

1, 5, 19, 95

How to compute the divisors of 190?

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 190 by each of the numbers from 1 to 190 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)

190 / 1 = 190 (the remainder is 0, so 1 is a divisor of 190)
190 / 2 = 95 (the remainder is 0, so 2 is a divisor of 190)
190 / 3 = 63.333333333333 (the remainder is 1, so 3 is not a divisor of 190)
...
190 / 189 = 1.005291005291 (the remainder is 1, so 189 is not a divisor of 190)
190 / 190 = 1 (the remainder is 0, so 190 is a divisor of 190)

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 190 (i.e. 13.78404875209). 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 $\frac{N}{d} = D$ )

190 / 1 = 190 (the remainder is 0, so 1 and 190 are divisors of 190)
190 / 2 = 95 (the remainder is 0, so 2 and 95 are divisors of 190)
190 / 3 = 63.333333333333 (the remainder is 1, so 3 is not a divisor of 190)
...
190 / 12 = 15.833333333333 (the remainder is 10, so 12 is not a divisor of 190)
190 / 13 = 14.615384615385 (the remainder is 8, so 13 is not a divisor of 190)