## Navigating Factors and Multiples in GRE: An Introduction

Welcome to a new chapter in your GRE quantitative section preparation! We, at MKSprep, know that a strong foundation in number properties, including factors and multiples, can be instrumental in boosting your GRE score. This guide will introduce you to the intriguing world of factors and multiples, helping you to decode their properties and apply them effectively in problem-solving.

Factors and multiples are fundamental concepts in number theory, often providing the key to unlocking complex GRE problems. Whether prime factorization or determining the least common multiple, these concepts weave into numerous GRE questions.

This six-part guide will take you through an immersive journey exploring factors and multiples. We’ll start with the basics, gradually delving into more complex concepts, formulas, and time-saving tricks. Additionally, we’ll share some effective test-taking and time-management strategies and conclude with examples of typical GRE questions.

By the end of this guide, we aim to equip you with a robust understanding of factors and multiples, enabling you to tackle related GRE questions with confidence and precision.

Keywords: GRE, Factors and Multiples, Number Properties, MKSprep.

## Understanding Factors and Multiples: Core Concepts for GRE

In the world of integers, factors and multiples hold a prominent place. Understanding their fundamental properties can be key to solving a variety of GRE quantitative questions. Let’s delve into these core concepts.

### Factors:

1. Definition: A factor of a number is an integer that divides that number evenly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
2. Properties of Factors:
• Every number is a factor of itself.
• 1 is a factor of every number.
• Every factor of a number is less than or equal to that number.
• The number of factors a number has depends on its prime factorization.

### Multiples:

1. Definition: A multiple of a number is the result of multiplying that number by an integer. For example, the multiples of 4 are 4, 8, 12, 16, and so on.
2. Properties of Multiples:
• Every number is a multiple of itself.
• Every number is a multiple of 1.
• Every multiple of a number is greater than or equal to that number.
• The set of multiples of a number is infinite.

These are the basic definitions and properties of factors and multiples. The next section delve into more complex concepts that build upon these foundations.

## Deep Dive into Factors and Multiples: Advanced Concepts for GRE

After understanding the basic properties of factors and multiples, let’s extend our knowledge to more advanced concepts. These concepts will further enhance your problem-solving skills for the GRE.

### Advanced Concepts Related to Factors:

1. Prime Factors: A prime factor of a number is a factor that is a prime number. Prime factorization involves expressing a number as a product of its prime factors.
2. Greatest Common Factor (GCF): The GCF of two or more numbers is the largest number that is a factor of all of them. Finding the GCF often involves the use of prime factorization.

### Advanced Concepts Related to Multiples:

1. Common Multiples: A common multiple of two or more numbers is a number that is a multiple of all of them.
2. Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number, a multiple of all of them. Like the GCF, finding the LCM often involves prime factorization.

Understanding these advanced concepts related to factors and multiples is crucial for handling complex GRE questions. In the next section, we’ll explore some useful formulas, tricks, and shortcuts related to factors and multiples.

## Factors and Multiples: Formulas, Tricks, and Shortcuts for the GRE

Armed with the fundamental and advanced concepts of factors and multiples, let’s now explore some crucial formulas, tricks, and shortcuts that can save you precious time on the GRE.

### Formulas:

1. Number of Factors: If the prime factorization of a number ‘N’ is expressed as a^p * b^q * c^r…, then the total number of factors of ‘N’ is (p+1)(q+1)(r+1)…
2. Least Common Multiple (LCM) and Greatest Common Factor (GCF): For any numbers ‘a’ and ‘b,’ the product of their LCM and GCF is equal to the product of the numbers. That is, LCM(a, b) * GCF(a, b) = a * b.

### Tricks and Shortcuts:

1. Shortcut for Finding GCF: Use the Euclidean algorithm – repeatedly subtract the smaller number from the larger until you get a remainder of 0. The last non-zero remainder is the GCF.
2. Shortcut for Finding LCM: Divide the product of the two numbers by their GCF.
3. Quick Prime Factorization: For numbers that are not too large, it’s often faster to find the prime factorization by dividing by prime numbers, starting from 2 and moving upwards.

With these tricks and shortcuts, you can tackle GRE questions involving factors and multiples more efficiently. In the next section, we’ll discuss some general problem-solving tips and time-management strategies.

## Conquering Factors and Multiples in GRE: Tips and Time Management Strategies

Understanding the concepts of factors and multiples is crucial, but knowing how to apply them effectively in the GRE is equally important. This section will share some practical tips and time management strategies.

### General Tips:

1. Understanding the Question: Take time to understand what the question is asking. Look for clues that suggest the use of factors or multiples.
2. Practice: Regular practice is essential. The more you practice, the more familiar you become with various types of questions and the more adept you become at applying the concepts of factors and multiples.
3. Use Scratch Paper: Don’t hesitate to use scratch paper for calculations and for laying out your thoughts. This can help prevent mistakes and provide a clear path to the solution.

### Time Management Strategies:

1. Skip Complex Questions: If a question appears too complex or time-consuming, skip it and return to it later if time permits.
2. Use Shortcuts: Use the tricks and shortcuts we discussed earlier whenever applicable. This can save valuable time.
3. Estimation: In some questions, rather than calculating the exact answer, it might be faster to estimate the answer and select the closest option.

Armed with these tips and strategies, you can approach GRE questions involving factors and multiples with increased confidence. In the next section, we will look at examples of these types of questions and walk through the solutions.

## Factors and Multiples in Action: GRE Question Types and Examples

In the final part of our guide, we’ll explore different types of GRE questions that involve factors and multiples, providing examples and solutions for each. This practical application will help cement your understanding of the concepts we’ve discussed.

### Quantitative Comparison Questions:

Example: Compare the two quantities.

Quantity A: The number of factors of 30. Quantity B: The number of multiples of 5 less than 30

Solution: The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Thus, Quantity A is 8. The multiples of 5 less than 30 are 5, 10, 15, 20, and 25. Thus, Quantity B is 5. Therefore, Quantity A is greater.

### Multiple-Choice Questions:

Example: What is the least common multiple of 8 and 12?

A) 24 B) 36 C) 48 D) 60

Solution: The multiples of 8 are 8, 16, 24, 32, 40, 48, 56… The multiples of 12 are 12, 24, 36, 48… The least common multiple is 24. So, the answer is A) 24.

### Numeric Entry Questions:

Example: If ‘n’ is a positive integer such that 5n is a factor of 100, what is the greatest possible value of n?

Solution: The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The largest factor that is a multiple of 5 is 100, so 5n = 100 and n = 20.

By working through these examples, you’ll get a sense of how factors and multiples appear in GRE questions and how to apply the concepts, formulas, and strategies we’ve discussed to solve these problems effectively.