Integers: Even and Odd - No Rules for Division

The rules of arithmetic using Odd and Even numbers are important in many GMAT integers questions. Let's see if we can formulate a rule for division of Even and Odd numbers.

$$\frac{Even}{Even}$$ $$is...$$

If there is one general rule that can be used for a specific case of division, it is this:

$$\frac{Odd}{Even}=$$$$Fraction$$

An odd number is not divisible by 2. An even number is divisible by 2, and will therefore be a multiple of 2. Therefore, an odd integer will never be divisible by an even integer, and the result will be a fraction.

That was a trick question to demonstrate a common misconception caused by linear thinking. Many people are impressed by the abundance of "even"s in the above expression and jump to the conclusion that the result must be even as well. However, try plugging in for an even number divided by an even number:

4 / 2 = 2 (an even result)

however,

6 / 2 = 3 (an odd result)

finally,

2 / 4 = 1/2 - (a fraction - not even an integer!)

So the answer to the above question is actually "I don't know - neither MUST be true". Which brings us to the following rule regarding division of Even and Odd numbers:

when dividing Even and Odd numbers, there are no rules. The result could be even, odd or even a fraction - depending on the numbers.

Some GMAT integers questions rely on this form of linear thinking to trick careless and hasty test-takers. Remember this rule, and do not be fooled.