A positive number N is divided into two unequal parts such that the difference of the cubes of the two parts is 133 times their difference. If the product of the two parts is 36, then what is the value of N?12131520

Let's denote the two parts as x and y. We know that x ≠ y and x*y = 36.

From the problem, we also know that x³ - y³ = 133*(x - y).

We can rewrite the equation x³ - y³ as (x - y)(x² + xy + y²).

So, we have (x - y)(x² + xy + y²) = 133*(x - y).

We can cancel out (x - y) from both sides, which gives us x² + xy + y² = 133.

We also know that (x + y)² = x² + 2xy + y².

Subtracting 2xy from both sides gives us x² + y² = (x + y)² - 2xy.

Substituting xy = 36 and x² + y² = 133 - xy into the equation gives us (x + y)² - 72 = 133 - 36.

Solving for (x + y)² gives us (x + y)² = 97.

Taking the square root of both sides gives us x + y = √97.

Since x and y are positive numbers, the value of N = x + y = √97 ≈ 9.85.

However, the problem asks for the value of N in the set {12, 13, 15, 20}.

Since 9.85 is not in this set, there seems to be a mistake in the problem. None of the provided options are correct.