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# Integers: Division and the Remainder Formula

In a division operation involving two integers, the dividend is even, the divisor is odd, and the remainder is zero. What must be true of the quotient?
Incorrect. [[snippet]] __Plug In__ an even dividend and an odd divisor—for example, $$\frac{2}{1}$$. The quotient is $$\frac{2}{1}=2$$. Therefore, this answer choice is not *must* be true. Eliminate and move on.
Incorrect. [[snippet]] __Plug In__ an even dividend and an odd divisor—for example, $$\frac{2}{1}$$. The quotient is $$\frac{2}{1}=2$$: even. Therefore, this answer choice is not *must* be true. Eliminate and move on.
Correct. [[snippet]] __Plug In__ an even dividend and an odd divisor—for example, $$\frac{2}{1}$$. The quotient is $$\frac{2}{1}=2$$. Answer choice A cannot be eliminated for this plug-in, but answer choices B, C, and D are proven wrong and are __POE'd__. Now, is the quotient always even? Since the remainder is 0, the quotient cannot be a fraction. You can try to __Plug In__ several more pairs of integers that satisfy the question stem (6 and 3 $$=$$ the quotient is even; 20 and 5 $$=$$ the quotient is still even). Under test conditions, now would be a good time to decide that the quotient is even. Choose answer choice A and move on. If you have the time, take a step back and consider the concepts behind this question: > $$\displaystyle \frac{\text{Dividend}}{\text{Divisor}} =$$ {color:red}Quotient{/color}. Manipulate the equation to get rid of the fraction: > Dividend $$=$$ {color:red}Quotient{/color} $$\times$$ Divisor. Since the dividend is even, the right side of the equation must also be even. And since the divisor is odd, the quotient must be even. If the quotient were odd, then the product of odd $$\times$$ odd would be *odd* as well. Something has to be even on the right side in order to get an even result.
Incorrect. [[snippet]] __Plug In__ an even dividend and an odd divisor—for example, $$\frac{2}{1}$$. The quotient is $$\frac{2}{1}=2$$. Therefore, this answer choice is not *must* be true. Eliminate and move on.
Incorrect. Oh, but there is. [[snippet]]
The quotient is even.
The quotient is odd.
The quotient is 1.
The quotient is 0.
There is not enough information to tell.