I OBM

Question

Show that the number of positive integer solutions to x_1 + 2^3 x_2 + 3^3 x_3 + + 10^3 x_10 = 3025 (*) equals the number of non-negative integer solutions to the equation y_1 + 2^3 y_2 + 3^3 y_3 + + 10^3 y_10 = 0 Hence show that (*) has a unique solution in positive integers and find it.

Accepted Answer

We have 1^3 + 2^3 + + 10^3 = 3025. Now x_i is a positive integer solution to (*) iff y_i = x_i - 1 are all non-negative and satisfy (y_1+1)+2^3(y_2+1)+3^3(y_3+1)++10^3(y_10+1) = 3025 and hence y_1+2^3y_2+3^3y_3++10^3y_10 = 0. But that clearly has the unique solution y_i = 0, so the unique solution to (*) is x_i = 1.