In this question, your task is to find integer solutions to the following equation. Ax$1+$Bx$2+$Cx$3+$Dx$4+$Ex$5=$S

where x$1,$ x$2,...,$ x$5$ are unknowns, and A$,$ B$,$ C$,$ D$,$ E$,$ S are given constants, all these constants are integers and can be positive, negative, or zero.

We want to use a computer algorithm to find the number of unique integer solutions to the above equation, with the constraint that each solution only includes unknowns satisfying the following condition:

$50<=$ xi $<=50$i in $\{1,2,3,4,5,6\}$

In the starter code A$3$Q$2.$java, you will implement the solve method that takes A$,$ B$,$ C$,$ D$,$ E$,$ S as input and returns

a long integer that is the number of unique integer solutions to the above equation.

A na $$ve algorithm would be to enumerate all possible $5-$tuples of $($x$1...$x$5)$ and check each tuple to see whether it satisfies the equation. The number of the $5-$tuples to check would be roughly $1005=1010,$ which will take the program a long time to process. So we would like to devise a faster algorithm with the help of a data structure we learned, namely, the hash table. The total number of steps taken by the algorithm should be much less than $1010.$

Requirements:

$$ Each call to your solve$()$ function must be able to return the correct answer within $5$ seconds for all test cases.

There are a few test cases provided in the main function of the starter code.

$$ You are not allowed to add any imports to the starter code, which means you cannot use the built$-$in Java classes such as java.util.HashMap and java.util.Hash