For what integer values of n is the fraction (4n-5) / (2n-1) a natural number?

Let’s select the whole part from the given fraction, and consider its fractional part, we get:

(4 * n – 5) / (2 * n – 1) = [2 * (2 * n – 1) – 3] / (2 * n – 1) = 2 – 3 / (2 * n – 1).

Consider the resulting new fraction 3 / (2 * n – 1), and determine the values of n, at which the fraction is an integer. This is an integer fraction for the following data n:

(2 * n – 1) * k = 3, or for the integer value of the fraction: (2 * n – 1) = 3; 2 * n = 1 + 3; 2 * n = 3; n = 4/2 = 2. Check: substitute the value n = 2, we get:

(4 * 2 – 5) / (2 * 2 – 1) = 3/3 = 1, found correctly.