Yes, both positive and negative integers exist.
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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.
Yes. It can also be negative in the numerator. Both positive and negative numbers (as well as zero) can be rational numbers. Both positive and negative numbers can be irrational numbers. Both positive and negative numbers (as well as zero) can be integers.
The quotient is what you get when you divide two numbers. If both numbers are positive, the quotient will be positive. If both numbers are negative, the quotient will be positive. If one number is positive and one number is negative, their quotient will be negative.
Start at 0 and go in both the negative and positive directions. So in the positive one we have 0,1,2,3,4,5,6,7,8,9,19, and keep going forever. Those are the non-negative integers. If we get rid of the o and leave all the others, we have the positive integers. Now take each positive integer, 1,2,3,4 etc and make it a negative number, so we have -1,-2,-3, etc, those are the negative integers. Or we can start at -1 and count toward negative infinity by going -1, -2,-3, -4, .... forever and those are the negative integers.
There are an infinite number of positive integers divisible by 3, 5, and 7. This is because there is no limit to numbers and they go on indefinitely in both a positive and a negative direction.