Alice, Bob, and Carol each think of an expression that is a fraction with 1 as a numerator and a constant integer times some power of x as the denominator. The simplest common denominator of Alice's and Bob's expressions is 4x2. The simplest common denominator Bob's and Carol's expressions is 12x3.. The simplest common denominator of Alice's and Carol's expressions is 6x3. Find all possible expressions that could be Carol's expression. The way I did it is just brute forcing through all the possible constant terms and then the powers of x that the Carol's expression could be. This way I arrived at the solutions [MATH]C = 1/(3x^3)[/MATH] and [MATH]C = 1/(6x^3)[/MATH]The downside of this method is that you can easily miss some combination especially as this numbers become bigger, so I wonder if there's some more elegant solution to this problem that assures you always find all of the answers. This is the problem from a 6-9 grades algebra textbook, although it is presented as an extra challenging one.