Integer Properties GMAT: Master the Fundamentals for a High Score

Key Takeaways
-GMAT integer properties questions test fundamental mathematical concepts such as divisibility, prime numbers, even-odd properties, and factors/multiples. Mastering these concepts is critical to successfully navigating the Quant section.
-Common question types include calculating the greatest common divisor (GCD), least common multiple (LCM), and analyzing prime factorization for solving divisibility problems.
-A common problem-solving example might involve determining the least common multiple (LCM) of two numbers by using prime factorizations.
-Understanding the rules of integers, divisibility, and prime factorization helps to quickly identify the solution path in both data sufficiency and problem-solving questions.

In the GMAT Quant section, integer properties are some of the most frequently tested topics, especially for those aiming for a 700+ score. Whether you’re tackling Problem Solving or Data Sufficiency questions, a solid understanding of integer properties GMAT is essential. These questions involve concepts like divisibility, prime numbers, even and odd integers, and greatest common divisors (GCD), all of which play a key role in determining your accuracy and speed.

In this comprehensive guide, we'll dive into the most important integer properties you need to know for the GMAT, practical strategies for solving integer-related questions, and detailed examples to help you excel in both problem-solving and data-sufficiency questions.

Understanding Key Integer Properties for GMAT Success

The GMAT Quant section requires a deep understanding of number theory, and integer properties form the backbone of many questions. Before we dive into strategies and sample questions, it’s important to clarify what integers are and explore their various properties.

What are Integers?

An integer is any whole number, positive or negative, including zero. Understanding what integers are and how they behave under different operations is fundamental to solving integer-based questions on the GMAT.

• Positive Integers: Numbers greater than 0 (e.g., 1, 2, 3...)
• Negative Integers: Numbers less than 0 (e.g., -1, -2, -3...)
• Zero (0): An integer that is neither positive nor negative, often used in operations that test boundaries and limits.

Even and Odd Integers

Even and odd integers are another critical property in GMAT integer questions. This classification determines how integers interact when they are added, subtracted, or multiplied.

• Even Integers: Any integer divisible by 2 (e.g., -4, -2, 0, 2, 4...)
• Odd Integers: Integers that are not divisible by 2 (e.g., -3, -1, 1, 3...)

Properties:

• Even + Even = Even
• Odd + Odd = Even
• Even + Odd = Odd
• Even × Even = Even
• Odd × Odd = Odd
• Even × Odd = Even

Prime Numbers and Prime Factorization

Prime numbers are positive integers greater than 1 that are divisible only by 1 and themselves. Prime numbers play a pivotal role in GMAT questions involving divisibility, factors, and multiples.

Key Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…

Composite numbers: Any integer that is not prime (e.g., 4, 6, 8, 9…) can be factored into smaller prime numbers.

Divisibility Rules

Divisibility rules help you quickly determine whether a number is divisible by another without performing long division. These rules are especially helpful when solving GMAT integer properties questions efficiently.

• Divisible by 2: The number ends in 0, 2, 4, 6, or 8.
• Divisible by 3: The sum of the digits is divisible by 3.
• Divisible by 5: The number ends in 0 or 5.
• Divisible by 9: The sum of the digits is divisible by 9.
• Divisible by 10: The number ends in 0.

Types of GMAT Questions on Integer Properties

Problem-Solving Questions

In Problem-Solving questions, you'll need to directly apply integer properties to solve for unknowns, find factors, or analyze divisibility. These questions typically involve algebraic expressions or word problems that require an understanding of integer properties.

Data Sufficiency Questions

Data Sufficiency (DS) questions test your ability to determine whether the given information is sufficient to solve a problem, often without fully solving it. Integer properties are commonly tested in DS questions because they allow for logical deductions about whether numbers are divisible, even, odd, prime, or composite.

Effective Strategies for Integer Properties GMAT Questions

Integer properties questions on the GMAT often test basic number concepts but require careful analysis and strategy to solve them efficiently. Below are three effective strategies to help you tackle these problems with confidence.

Strategy 1: Process of Elimination

One of the most effective and time-saving techniques for GMAT integer questions is the process of elimination. By recognizing the key integer properties mentioned in the question, you can quickly eliminate answer choices that don't meet the criteria. For example, if a problem asks for an odd integer as the answer, you can immediately rule out any even-numbered options, narrowing down the possibilities.

This strategy is particularly helpful in multiple-choice questions where scanning through the answer options and applying basic rules (like divisibility or even-odd properties) can lead you closer to the solution without needing to perform time-consuming calculations. For example, if a problem involves divisibility by 3, eliminate any answer choices that do not meet this criterion. Using this approach reduces cognitive load and increases efficiency under time pressure.

Strategy 2: Prime Factorization

Prime factorization is a powerful tool for solving a variety of GMAT integer problems, especially those dealing with divisibility, greatest common divisors (GCD), and least common multiples (LCM). By breaking a number down into its prime factors, you gain a clearer understanding of its properties and how it interacts with other integers. This strategy is essential when determining common divisors or multiples of two numbers.

For instance, consider a problem asking for the least common multiple (LCM) of 18 and 24. First, break down each number into its prime factors:

• 18 = 2 × 3²
• 24 = 2³ × 3

The LCM is determined by taking the highest powers of all primes involved. In this case, the highest powers are 2³ and 3². Therefore, the LCM is:

• 2³ × 3² = 72

Prime factorization simplifies otherwise complex calculations and ensures accuracy, especially in problems involving larger numbers.

Strategy 3: Plugging In Numbers

When faced with an abstract question or a complex algebraic expression involving integers, a useful strategy is to plug in small numbers to test different scenarios. This method is especially effective for data-sufficiency questions where you need to determine whether a statement is sufficient to answer the question.

For example, if a question asks whether an expression is divisible by a particular integer, try substituting small values like 1, -1, 2, or -2 into the variables. This technique allows you to observe how the integer properties change with different inputs, helping you evaluate whether a condition holds true under all circumstances.

Consider the following example:

• If aaa and bbb are integers and a2−b2a^2 - b^2a2−b2 is divisible by 4, what must be true?

The expression a2−b2a^2 - b^2a2−b2 can be factored as (a−b)(a+b)(a - b)(a + b)(a−b)(a+b). For the product to be divisible by 4, both a−ba - ba−b and a+ba + ba+b must either be even, or one of them must be divisible by 4. By plugging in small values for aaa and bbb, you can verify this pattern, making the question more approachable.

Sample GMAT Integer Property Questions

Problem Solving Example 1

What is the least common multiple (LCM) of 18 and 24?

Solution:

Prime factorization of 18: 2 × 3²
Prime factorization of 24: 2³ × 3
The LCM is the product of the highest powers of all primes: 2³ × 3² = 72.

Problem Solving Example 2

If a and b are integers and a² - b² is divisible by 4, which of the following must be true?

Solution:

a² - b² = (a - b)(a + b). For the product to be divisible by 4, both a - b and a + b must either be even, or one must be divisible by 4.

Practice GMAT Integer Property Problems

Mastering GMAT integer property questions requires practice with real examples to hone your understanding of divisibility, prime factorization, greatest common divisors (GCD), and least common multiples (LCM). Below, we elaborate on two essential practice problems, breaking down the solution steps to help you develop a clear strategy for solving similar GMAT integer questions.

Practice Question 1

Question: What is the greatest common divisor (GCD) of 48 and 64?

Solution:

Step 1: Prime Factorization

The first step in finding the greatest common divisor (GCD) is to break each number down into its prime factors.

• Prime factorization of 48: 48 = 2⁴ × 3
  This is because 48 can be factored as: 48 ÷ 2 = 24, 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3, leaving the prime factorization as 2⁴ × 3.
• Prime factorization of 64: 64 = 2⁶
  Since 64 is a power of 2, it can be written as 2⁶.

Step 2: Identify the Common Factors

• To find the GCD, you need to identify the prime factors common to both numbers. In this case, the only common prime factor is 2.
• Now, take the lowest power of the common prime factor. Between 2⁴ and 2⁶, the lowest power is 2⁴.

Step 3: Calculate the GCD

The GCD is the product of the lowest powers of all common prime factors. Here, it’s 2⁴, which equals 16.

Conclusion: The greatest common divisor (GCD) of 48 and 64 is 16. By breaking down both numbers into their prime factors, you can systematically identify the GCD, a crucial skill for solving GMAT integer property problems.

Practice Question 2

Question: If x is an integer and x² is divisible by 12, which of the following must be true?

Solution:

Step 1: Understand the Problem

The question states that x², the square of an integer, is divisible by 12. We need to determine what must be true about x, given that x² is divisible by 12.

Step 2: Prime Factorization of 12

• First, we factor 12 into its prime factors to understand the divisibility requirements:
• 12 = 2² × 3
  This means x² must be divisible by 2² × 3, or both the square of 2 and 3.

Step 3: Determine the Properties of x²

• Since x² is divisible by 2² and 3, x itself must be divisible by at least one factor of 2 (because squaring 2 results in 2²) and by 3 (since squaring 3 results in divisibility by 3).
• Therefore, x must be divisible by both 2 and 3 to ensure that x² is divisible by 2² × 3. In other words, x must be divisible by the least common multiple of 2 and 3.

Step 4: Conclusion

• The least common multiple (LCM) of 2 and 3 is 6. Therefore, x must be divisible by 6 for x² to be divisible by 12.
• Conclusion: x must be divisible by 6.

Explanation: This problem demonstrates the use of prime factorization to determine the divisibility of an integer’s square. The key idea here is that for x² to be divisible by 12, x itself must contain all the necessary prime factors (in this case, 2 and 3). The lowest integer divisible by both 2 and 3 is 6, meaning x must be divisible by 6 to satisfy the condition.

GMAT Test-Day Tips for Integer Properties

The GMAT is as much a test of strategy as it is of knowledge. When it comes to integer property questions, being well-prepared with the right techniques can save you valuable time and boost your overall score. Here are some essential tips to help you master integer property questions on test day:

Memorize Divisibility Rules

Memorizing basic divisibility rules can greatly speed up your problem-solving process. These rules tell you when one number can be evenly divided by another without leaving a remainder. Here are some key divisibility rules to keep in mind:

• A number is divisible by 2 if its last digit is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 6 if it is divisible by both 2 and 3.

These quick checks are particularly useful when you need to eliminate incorrect answer choices in a matter of seconds. For example, in a multiple-choice question asking whether a number is divisible by 6, applying the rules for 2 and 3 can allow you to instantly narrow down the options. Memorizing and practicing these rules beforehand will help you solve divisibility questions faster and with greater accuracy.

Be Strategic in Data Sufficiency

In GMAT Data Sufficiency (DS) questions, your goal is not to find the actual solution but to determine whether the given information is sufficient to answer the question. When it comes to integer property questions in DS, resist the temptation to fully solve the problem. Instead, focus on analyzing whether each statement provides enough information to arrive at a definitive answer.

For example, if a question asks whether a number is divisible by 3, and one of the statements tells you that the sum of the digits of the number is divisible by 3, you can immediately conclude that this statement is sufficient. Don’t waste time performing further calculations. Your focus should always be on assessing sufficiency, not solving the entire problem, which is a common mistake test-takers make.

Being strategic with DS questions can save you considerable time, which you can then allocate to more challenging problems.

Practice Factorization

Prime factorization is one of the most valuable tools for solving integer property questions, especially those involving divisibility, greatest common divisor (GCD), and least common multiple (LCM). Factorizing numbers into their prime components helps you understand how those numbers relate to each other in terms of their factors and multiples.

For example, if a question asks for the LCM of two numbers, first break each number down into its prime factors. Then, take the highest powers of all the primes involved to find the LCM. Similarly, to find the GCD, you’ll take the lowest powers of the common prime factors.

Practice factorizing different integers and applying this technique to GCD and LCM questions so that it becomes second nature during the exam. The more you practice, the faster and more intuitive the process will become, allowing you to solve questions efficiently without second-guessing.

Manage Time Efficiently

Time management is crucial in the GMAT, especially in the Quantitative section. Integer property questions, while foundational, can sometimes be tricky or time-consuming if you don’t use the right shortcuts. Use the following strategies to manage your time effectively:

• Eliminate Incorrect Answers: As mentioned earlier, the process of elimination is one of the fastest ways to narrow down your options. Quickly eliminate any choices that don’t fit the criteria given in the question (e.g., even numbers, divisible by specific primes).
• Use Prime Factorization: When faced with questions about divisibility, GCD, or LCM, use prime factorization as your go-to strategy. It simplifies the problem and often leads directly to the correct answer.
• Know When to Move On: If a question seems too complicated or time-consuming, mark it and move on. Spend more time on questions you’re confident you can solve quickly. You can always come back to more challenging questions if time permits.

Remember, GMAT integer questions are designed to test your knowledge and efficiency. By using shortcuts like elimination and prime factorization, you can manage your time effectively and avoid getting stuck on any one question.

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Conclusion

Understanding integer properties is a must for excelling in the GMAT Quant section. By mastering even/odd integers, divisibility, prime factorization, and strategies like elimination and plugging in numbers, you'll not only answer integer property questions correctly but also improve your speed and accuracy. With consistent practice, these concepts will become second nature, allowing you to score higher and achieve your GMAT goals.