A closed-form solution is obtained for the mode shape and the buckling load of an axially graded column that is clamped at one end and subjected to a concentrated load at the other. A semi-inverse method is employed to obtain the spatial distribution of the elastic modulus variation. A remarkable conclusion is reached on the existence of three columns with different elastic modulus variations, with the same analytical expression of the buckling load. The mode shape of one of the columns is represented by a rational expression while the other two involve irrational numbers.