One or both of the numbers must be zero.
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The product of two integers will be: * Zero, if one factor, or both, are zero. * Positive, if both factors have the same sign (both positive, or both negative) * Negative, if the two factors have different signs. Actually, these rules apply to all real numbers, not just to integers.
Due to carries, in the multiplication a zero can change to a non-zero and vice versa.
'Product' is another way of saying 'multiply the two numbers together'.
I have no idea that the product could be musical! If the two numbers have the same sign, the product is positive. If two numbers have different signs the product is negative.