An easy proof of Ramanujan's famous congruences $p(5m+4)\equiv 0 \equiv \tau(5m+5) \pmod 5$

Abstract

We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } \tau(5n+5) \equiv 0 \pmod 5.$$ The proof only requires a limiting case of Jacobi's triple product, a result that Ramanujan knew well, and a technique which Ramanujan used himself to compute values of $\tau(n)$.