Problem $1(12$ pts$.,2$ pts$.$ each$)$: Condition Strength

Order conditions from the weakest to the strongest by showing the longest implication chain

of the form A $$ B $$ C $$ D $...($where A$,$ B$,$ C$,$ D$,...$ are given conditions, $$ is implication$).$

If total ordering by strength cannot be established, show a partial ordering. Prove by showing all

necessary implications. If any implication is false, provide a counterexample. You must show all

work in order to get full credit. Unless otherwise stated, assume all referenced variables are defined

as integers