Question 4
Show that the number of positive integer solutions to
$$
x_1 + 2^3 x_2 + 3^3 x_3 + \cdots + 10^3 x_{10} = 3025 \quad (*)
$$
equals the number of non-negative integer solutions to the equation
$$
y_1 + 2^3 y_2 + 3^3 y_3 + \cdots + 10^3 y_{10} = 0
$$
Hence show that (*) has a unique solution in positive integers and find it.
$$
x_1 + 2^3 x_2 + 3^3 x_3 + \cdots + 10^3 x_{10} = 3025 \quad (*)
$$
equals the number of non-negative integer solutions to the equation
$$
y_1 + 2^3 y_2 + 3^3 y_3 + \cdots + 10^3 y_{10} = 0
$$
Hence show that (*) has a unique solution in positive integers and find it.