Final answer:
A counterexample to the statement that the quotient of two integers is an integer is when 1 is divided by 2, resulting in 1/2, which is not an integer. This proves that it's false to say the quotient will always be an integer. Integers do not include fractions, and dividing two integers that don't divide evenly will result in a fraction or decimal.
Step-by-step explanation:
A counterexample to the statement 'The quotient of two integers is an integer' can be easily provided. For example, consider the integers 1 and 2. When we divide 1 by 2, the result is ½, which is clearly not an integer, but a fraction. Therefore, this is a counterexample to the original statement, proving that it's not always true that the quotient of two integers is an integer.
Integers can be positive, negative, or zero, but they do not include fractions or decimals. When two integers that do not divide evenly are divided, the result is a fraction or a decimal, and not an integer. This is consistent with the rules of mathematics which apply universally; that is, if the same operation is performed on both sides, the expression remains an equality.
In contrast to a fraction that has the same quantity in both the numerator and denominator, which equals 1, dividing integers that don't result in a whole number clearly demonstrates that the result can be a non-integer.