Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 29 / 2 / iv

Use the standard theorem on reduction of torsion points on an elliptic curve: at a prime of good reduction, reduction is injective on torsion whose order is prime to that prime. Reduction at $2$ implies that every odd-primary part of the rational torsion subgroup has order dividing $3$. Thus its only possible odd prime is $3$, and its $3$-primary part has order at most $3$. Reduction at $5$ injects the $2$-primary part into a group of order $9$, so that part is trivial. No other primary part can occur.

Consequently the torsion subgroup has order at most three. Since $P_1$ has exact order three,This argument uses only prime-to-residue-characteristic injectivity, and therefore avoids an unjustified claim of full torsion injectivity at the prime $2$.