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1 is not composite, nor prime, because it is a "unit", i.e. it has only one factor, itself. Multiplying it by itself (or any other unit, such as the imaginary unit i) gives a unit (1, when we multiply it by itself.). This is actually a property of all units.

0 is not a composite (or prime) number because it cannot be expressed by multiplying prime numbers (one of the factors has to always be 0), and because it has infinitely many factors (any number times 0 is 0).

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Q: Why one and zero are not composite numbers?
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