The product of twenty-one negative integers is negative.
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odd number of factors ---> negative product
even number of factors ---> positive product
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Each 2 negative factors multiply to positive.
I you have an even number of factors, you can group them two by two. Each two negative factors has a positive product, and the product of all the positive factors is positive.
If you have an odd number of negative factors, then you can group all the factors two by two, but there will be one negative factor left over. All the negative factors grouped two by two will become positive products, but the one negative factor left over will make the final product negative.