It frequently happens that “integers” for test-taker are associated only with positive whole numbers, but actually they also include positive and negative numbers as well as 0. On the GRE there are a lot of problems involving integers and you should be familiar with their properties, since most of these problems are underrated by test-takers and have many hidden pitfalls.
Negative and Positive
Numbers can be either positive or negative (except the number 0, which is neither). Negative numbers are all to the left of zero. Positive numbers are all to the right of zero:
Note that a variable (such as x) can have either a positive or a negative value, unless there is evidence otherwise.
Note: On GRE the variable x is not necessarily positive, nor is –x necessarily negative.
Properties of Negative and Positive
By now you probably know how to add or subtract positive/negative numbers, so as a reminder subtracting negative from positive: 3 – (–2) = 3 + 2 = 5. This is very easy to miss, especially if you work with parentheses: 7 – (12 – 9) = 7 – 12 + 9 = 4.
Now, when you multiply or divide two numbers, positive or negative, follow one simple rule: If Signs are the Same, the answer’s poSitive but if Not, the answer is Negative:
This principle can be extended to multiplication and division by more than two numbers. For example, if 3 numbers are multiplied together, the result will be positive if there are NO negative numbers, or TWO negative numbers. The result will be negative if there is ONE or THREE negative numbers.
We can summarize this pattern as follows. When you multiply or divide a group of nonzero numbers, the result will be positive if you have an EVEN number of negative numbers. The result will be negative if you have an ODD number of negative numbers.
The absolute value of a number answers this question: How far away is the number from 0 on the number line? For example, the number 5 is exactly 5 units away from 0, so the absolute value of 5 equals 5. Mathematically, we write this using the symbol for absolute value: |5| = 5.
To find the absolute value of –5, –5 is also exactly 5 units away from 0. Thus, the absolute value of –5 equals 5, or, in mathematical symbols, |–5| = 5.
Notice that absolute value is always positive, because it disregards the direction (positive or negative) from which the number approaches 0 on the number line. Note: |0| = 0 is the smallest possible absolute value.
Even and Odd
Even numbers are integers that are divisible by 2. Odd numbers are integers that are not divisible by 2. All integers are either even or odd.
|Even||is divisible by 2||2k||0, 2, 4, 6, 8|
|Odd||is not divisible by 2||2k + 1||1, 3, 5, 7, 9|
Note: 0 is even! Negative integers are also either even or odd.
Properties of Odd and Even
The GRE tests your knowledge of how odd and even numbers combine through addition, subtraction, multiplication, and division. Rules for adding, subtracting, multiplying and dividing odd and even numbers can be derived by simply picking numbers and testing them out. While this is certainly a valid strategy, it also pays to memorize the following rules for operating with odds and evens, as they are extremely useful for certain GRE math questions.
Note: Same rules for subtraction and division respectively.
Efficiently refreshing your knowledge base of math is one of the more important aspects of any quality GRE preparation. For instance, the order of operations is certainly something of which almost all of us had a decent understanding at one time or another. And the first thing which comes to mind when talking about operations order is a common mnemonic PEMDAS: “Please Excuse My Dear Aunt Sally.” Let’s further examine this prioritization process:
- Parentheses: first do whatever appears in parentheses, following PEMDAS within the parentheses if necessary.
- Exponents: next evaluate all terms with exponents.
- Multiplication and Division: then do all multiplications and divisions in order from left to right.
- Addition and Subtraction: finally, do all additions and subtractions in order from left to right.