  We study Fisher markets that admit equilibria wherein each good is integrally
assigned to some agent. While strong existence and computational guarantees are
known for equilibria of Fisher markets with additive valuations, such
equilibria, in general, assign goods fractionally to agents. Hence, Fisher
markets are not directly applicable in the context of indivisible goods. In
this work we show that one can always bypass this hurdle and, up to a bounded
change in agents' budgets, obtain markets that admit an integral equilibrium.
We refer to such markets as pure markets and show that, for any given Fisher
market (with additive valuations), one can efficiently compute a "near-by,"
pure market with an accompanying integral equilibrium.
  Our work on pure markets leads to novel algorithmic results for fair division
of indivisible goods. Prior work in discrete fair division has shown that,
under additive valuations, there always exist allocations that simultaneously
achieve the seemingly incompatible properties of fairness and efficiency; here
fairness refers to envy-freeness up to one good (EF1) and efficiency
corresponds to Pareto efficiency. However, polynomial-time algorithms are not
known for finding such allocations. Considering relaxations of proportionality
and EF1, respectively, as our notions of fairness, we show that fair and Pareto
efficient allocations can be computed in strongly polynomial time.
