Table of Contents
Definitions
An integer is any whole number. It can be greater than or less than zero.
Example
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4
A prime number is an integer greater than 1 whose only factors are 1 and itself.
Prime numbers show up very frequently, particularly on more difficult quantitative questions. It’s helpful to memorize the first ten prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Two important facts about prime numbers deserve special mention because they are commonly tested in more advanced number property questions.
- The number 1 is not a prime number.
- The number 2 is the only even prime number.
Even numbers are divisible by the number 2.
0, 2, 4, 6, 8, 10, 12, 14
Odd numbers are any numbers not divisible by 2.
1, 3, 5, 7, 9, 11, 13
The GMAT will often test your ability to distinguish between even and odd numbers, adding difficulty to problems by inserting complex operations. Keep this simple rule in mind – any number divisible by 2 is even, including the number 0.
Factors and Multiples
Multiples of a number are the result of multiplying a number by another whole number; multiples must be equal to or larger than the integer. As a result, there are an infinite number of multiples.
Example
The example above is a multiplication table that shows the first 12 multiples of 6. The row starting with the variable x displays the integers that 6 is multiplied by to obtain the multiple.
A factor of a number will divide that number evenly, without any remainder. Factors must be able to divide into the integer; factors must also be less than or equal to the integer. As a result, there are a finite number of factors. A good technique to remember the difference between factors and multiples is that multiples are big (bigger than the original number), and factors are small (smaller than the original number).
Example
What are the factors of 12?
The factors of 12 are: 1, 2, 3, 4, 6, 12
Divisibility of Factors
An important consequence of factors is that if two numbers are both divisible by a given number x, the difference, sum, and product will also be divisible by the number x.
Example
25 and 10 are both divisible by 5.
Products of Factors
The product of two integers x and y is divisible by each factor of x and each factor of y.
Example
The product of 4 and 21 (which is 84) is divisible by all the factors of 4 and all the factors of 21.
Factors of Consecutive Integers
The GMAT absolutely loves to test your knowledge of consecutive integer properties. The following is a list of rules often tested on the GMAT. While these rules are all very intuitive, time constraints on the test do not allow us to rationalize and prove these rules out. Memorize them now and save precious minutes on the exam.
- A set of n consecutive integers must have a multiple of n among its members.
Example
4, 5, 6, 7, 8.Of the 5 consecutive integers listed above, one number must be a multiple of 5.
- For any set of consecutive integers with an odd # of terms, the sum of all the integers is always a multiple of the # of terms. This rule does not apply to the sum of a consecutive set with an even number of terms
Example
Is the sum of three consecutive integers divisible by 3?
Since 3 can be factored out, the sum of three consecutive integers will always be divisible by 3.
Is the sum of four consecutive integers divisible by 4?
However, this doesn’t work for consecutive numbers with an even number of terms.
- A single even integer in a consecutive series means the product of the series is divisible by 2. Two even integers in a consecutive series mean the product of the series is divisible by 4.
Positive and Negative Integers
Similar to even and odd numbers, the GMAT will also test your knowledge of positive and negative numbers when the numbers are subjected to complex transformations.
The following list contains specific rules about number properties. Most of them are fairly intuitive, but these rules may not be completely obvious under the stress of the exam. Make sure you memorize this list.
Fact: zero is neither positive nor negative.
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