Integers are the "counting numbers" and their negative counterparts, and zero. Opposite integers are the pairs of integers that have the same absolute value, or, in other words, are the same distance from zero. 10 and -10 are opposite integers. 43 and -43 are opposite integers. It's just that simple.
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Positive and negative integers are opposite each other.
if the signs are the same you must add its opposite.
Quotient positive: Both integers have the same sign: both positive or both negative. Quotient zero: The first integer is 0. Quotient negative: The integers have opposite signs: one positive and one negative.
In one sense, the opposite of the GCF could be the least common factor, but the least common factor of any set of positive integers is 1, so it's not very interesting.
A counting number is the numbers you lear as a little kid, counting numbers are one and up. Integers include the counting numbers, 0, and the opposite (negative) of counting numbers. So yes, a counting number or the opposite of a counting number is an integer.