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Fundamental Math for the GMAT: Properties of Integers

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will cover some fundamental properties of integers that will help you to refresh your knowledge of the properties of integers while preparing for the GMAT. We will also look at applications of these properties for a bit of practice.

Integers

Suppose you decide to take a job as a dog walker. As you're planning your week, you determine how many dogs you will be walking each day. This is where things get mathematical! You see, the number of dogs that you walk each day is a special type of number called an integer. Integers are whole numbers and their negatives.

  • Integers = {…, -3, -2, -1, 0, 1, 2, 3,…}

Since it is impossible to walk a fraction of a dog, the number of dogs that you walk on a given day will always be an integer.


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Now that we're clear about the definition of integers, let's talk about their properties.

Properties of Integers

Because the properties of integers are one of the most commonly tested concepts on the GMAT, they are a very important topic to study while preparing! Let's explore some of these properties.

  • Even and odd integers: The even integers are multiples of 2 (2k, where k is an integer), and the odd integers are even integers plus 1 (2k + 1, where k is an integer). Because of these definitions, the integers on the number line in consecutive order alternate as even and odd.


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  • Greatest common factor (GCF) and least common multiple (LCM) of two integers: The GCF of two integers is the largest number that divides into both integers evenly, and the LCM of two integers is the smallest number that they both divide into evenly.


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  • Quotient and remainder when dividing integers: When we divide integers (a/b), we end up with the number of times b fits into a and any left over amount. We call the number of times b fits into a the quotient, and we call the left over number the remainder.


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  • Prime numbers: Prime numbers are integers greater than 1 that are only divisible by 1 and themselves. The list of prime numbers is infinitely long and has changing patterns.


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  • Consecutive Integers: A list of consecutive integers are integers that start at an integer and go up by 1 to get to the next integer. In general, they have the form n, n + 1, n + 2, n + 3, n + 4, …


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  • Special properties of 1 and of 0: We can multiply or divide any integer by 1 and we get the same integer back out. If we add or subtract 0 to any integer, we get that integer back out. If we multiply any integer by 0, we get 0. Lastly, it is impossible to divide a number by 0, so division by 0 is always undefined.


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Of course, there are more properties of integers, but we only have so much room in this lesson! Knowing these properties is just as important as being able to apply them to problems on the GMAT, so let's consider a few examples.

Examples

The fun thing about properties of integers is that the problems involving them are kind of like solving puzzles, and who doesn't like a good puzzle? We want to understand the problem, and then figure out how to use our properties to answer them. Let's give it a try!

1. If a and b are odd integers, is the expression a + b - 1 even or odd?

Answer: Let's look back at our properties of combining even and odd integers. Notice that if we add two odd integers, we get an even integer. Therefore, we have that a + b is even. Now, we consider the fact that the even and odd integers alternate, so if we go one up or one down from a + b, we will get an odd integer. Therefore, a + b - 1 must be odd.

2. What is the sum of the prime numbers that are greater than 20, but less than 30?

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