Everyone else has answered it well, but I’d like to add that the notes are only the same if it’s A natural minor. There is also harmonic and melodic minor scale, which have a raised 7th note, or raised 6th and 7th.
So a piece in A minor could be identified by the use of F# and/or G#. Neither of these are likely to be in a piece in C major (though it’s not impossible).
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