We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } τ(5n+5) \equiv 0 \pmod 5.$$ 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 $τ(n)$.
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