The claim that silver is needed in order to kill a werewolf may have a number of origins.
German literature in 1640 mentions people killing were wolf with silver bullets. Additionally, by the english folklore shape shifters are vulnerable to silver. And a werewolf is, indeed, a shapeshifter.
From the year 1935 and onwards novelists claimed that the Beast of Gévaudan, a great wolf-like man eating creature, was shot and killed by a silver bullet.

One more origin is actually scientific. Silver, unlike many other common metals, reacts with the solfur in the air creating silver sulfide. When a werewolf gets stabbed or shot by a silver weapon, the sulfur enters it's bloodstream, blocks blood vessels and poisons cells. This makes the almost invincilbe animal die.

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