Let y be an element of the set A = {1, 2, 3, 5, 6, 10, 15, 30} and x1, x2, x3 be integers such that x1x2x3 = y, then the number of positive integral solutions of x1x2x3 = y is

Number of solutions of the given equation is the same as the number of solutions of the equationx1x2x3x4=30=2×3×5Here, x4 is there because if x1x2x3 = 15, then x4 = 2 and if x1x2x3 = 5, then x4 = 6, etc.x4 is in fact a dummy variable.Each of 2, 3 and 5 will be a factor of exactly one of x1, x2, x3, x4 in 4 ways.∴ Required number = 43 = 64