find the number of tuples of positive integers

Question

find the number of tuples (a,b,c,d) of positive integers

\begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array}

answer should be one of

0 , 1 , 2 , 4

Answer 1

$b^2,d^2$ are squares, so $a^3,c^3$ are squares, so $a,c$ are squares. The only positive squares with difference 64 are $17^2-15^2$ and $10^2-6^2$. So there are just two solutions for $(a,c)=(36,100),(225,289)$. It is easy to see that in each case there is one solution for $(b,d)$. So the total number of solutions is 2.