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Without the requirement that ''f''(1) = 1, one could still have ''f''(1) = 0, but then ''f''(''a'') = 0 for all positive integers ''a'', so this is not a very strong restriction.
The definition above can be rephrased using the language of algebra: A completely multiplicative function is an endomorphism of the monoid <math>(\mathbb Z^+,\cdot)</math>, that is, the positive integers under multiplication.
==Examples==
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