$($a$)$ Show that ILP is NP$-$complete.

Hint: You may use the following: Let N be the length of the encoding of A and b$.($Example of an encoding: all entries from A and b are encoded in binary and written consecutively with a fitting separator.$)$ Then there is a polynomial p such that if there is an $xin{Z}^n$ with Ax b$,$ then there is also an $xin{Z}^n$ with Ax b and for all j$.$

$($b$)$ RLP is defined similar to ILP: The only difference is that the question is if there is an $xin{Q}^n$ with Ax b$,$ instead of $xin{Z}^n.$ It is known that RLP $in$ P$.$ Why doesn$$t the previous reduction show P $=$ NP or in other words: why does your reduction from the previous exercise noShow that ILP remains NP$-$complete even if we only allow equations instead of inequalities, i$.$e$.$ we require Ax $=$ b$.$ Additionally we require x $>=0.$t work for RLP$?$

$(c)$ Show that ILP remains NP$-$complete even if we only allow equations instead of inequalities, i$.$e$.$ we require Ax $=$ b$.$ Additionally we require x $>=0.$