Negative numbers can be classified as either prime or composite because they still have their absolute value
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The number 1 is neither prime nor composite.
A prime number has only 2 factors which are 1 and itself. Composite numbers are everything else except 1 and 0. 1 and 0 are neither prime, nor composite. -2 is composite.
The product of two prime numbers will be composite.
A composite number is defined as a "positive integer with a positive divisor other than one and itself". So therefore, while negative numbers can have factors in the same way as positive numbers, none is considered prime.
Products of prime numbers are composite numbers.