## GRE Unit digit (Number Property)

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## GRE Unit digit

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## Mastering Unit Digits: An Introduction to the GRE Quantitative Section

Introduction: Welcome to Max Score Academy’s GRE blog series, dedicated to helping you excel in the Quantitative section of the GRE. This eight-part series will delve into the fascinating world of unit digits, a topic that often appears in GRE math questions. Understanding unit digits is crucial for solving problems quickly and accurately, allowing you to maximize your score on the exam.

## Part 1: Introduction

This introductory article will overview unit digits and their significance in GRE mathematics. We’ll explore why unit digits matter and how they can simplify complex calculations, giving you an edge in solving quantitative problems efficiently.

What are Unit Digits? Unit digits, also known as ones digits, are the rightmost digits of a number when written in base 10. For example, in the number 524, the unit digit is 4. Similarly, in the number 729, the unit digit is 9.

**Significance in GRE Math:** Unit digits play a crucial role in GRE math questions, particularly in topics such as number properties, algebra, and arithmetic. They often appear in questions that involve simplifying expressions, determining remainders, or identifying patterns in sequences.

Simplifying Calculations: Understanding unit digits allows you to simplify calculations, making them faster and more manageable. By focusing on the unit digits rather than the entire numbers, you can often eliminate unnecessary steps and quickly narrow down answer choices.

Identifying Patterns and Properties: Unit digits possess unique patterns and properties that can be leveraged to solve complex problems. These patterns can aid in determining divisibility, identifying cyclicity, or recognizing repeating sequences.

Example: Let’s consider the multiplication of two numbers: 36 and 48. Instead of multiplying the entire numbers, focusing on the unit digits alone reveals that the unit digit of the product will be 8 (from 6 × 8). This simple observation eliminates the need for extensive multiplication and speeds up problem-solving.

**Strategies for Utilizing Unit Digits:** Throughout this series, we will provide you with various strategies, formulas, and techniques to effectively utilize unit digits in GRE questions. These approaches will enable you to approach problems, optimize time management, and improve accuracy confidently.

Conclusion: In this first part of our eight-part series on unit digits, we have introduced the concept of unit digits and highlighted their significance in GRE math questions. Understanding unit digits empowers you to simplify calculations, identify patterns, and solve problems efficiently. In the upcoming parts, we will explore the concept of unit digits in greater detail, including formulas, tricks, techniques, general tips, time management strategies, and types of questions with examples.

Stay tuned for Part 2: Concept, where we will delve deeper into the principles underlying unit digits and their application in GRE mathematics. Remember, mastering unit digits will enhance your problem-solving abilities and contribute to your overall success on the GRE Quantitative section.

Max Score Academy is dedicated to providing comprehensive GRE preparation, and we are excited to guide you through this essential topic.

## Part 2: Concept – Unveiling the Principles of Unit Digits

Welcome back to Max Score Academy’s GRE blog series on unit digits. This second part will delve deeper into the principles underlying unit digits and explore their application in GRE mathematics. By understanding these concepts, you will gain a solid foundation for tackling various types of questions related to unit digits.

Understanding the Unit Digit Cycle: One of the key aspects of unit digits is their cyclical nature. Each digit from 0 to 9 follows a specific pattern when raised to different powers. This pattern, known as the unit digit cycle, allows us to predict the unit digits of numbers raised to different exponents.

**Unit Digit Cycle: **0: 0 1: 1 2: 2, 4, 8, 6 (repeats every 4 powers) 3: 3, 9, 7, 1 (repeats every 4 powers) 4: 4, 6 (repeats every 2 powers) 5: 5 6: 6 7: 7, 9, 3, 1 (repeats every 4 powers) 8: 8, 4, 2, 6 (repeats every 4 powers) 9: 9, 1 (repeats every 2 powers)

Example: Let’s consider the number 7 raised to different powers: 7^{1}, 7^{2}, 7^{3}, and so on. By referring to the unit digit cycle for 7, we can determine that the unit digits follow the pattern 7, 9, 3, 1. Therefore, 7^1 has a unit digit of 7, 7^2 has a unit digit of 9, 7^3 has a unit digit of 3, and so on.

Using the Unit Digit Cycle in Problem Solving: The unit digit cycle is invaluable when dealing with questions involving exponents, remainders, and divisibility. By identifying the unit digit of a base number and considering the exponent’s position in the cycle, you can quickly determine the unit digit of the result.

Example: Suppose you need to find the unit digit of 2^{17}. By referring to the unit digit cycle for 2, you know that the unit digits repeat every 4 powers (2, 4, 8, 6). Since 17 is not a multiple of 4, you can conclude that the unit digit of 2^{17 }will be the same as the unit digit of 2^{1}, which is 2.

**Strategies for Applying Unit Digit Principles:** Throughout this series, we will equip you with various strategies for utilizing unit digit principles effectively. These strategies will help you quickly identify patterns, eliminate answer choices, and solve problems efficiently. Incorporating these techniques into your GRE preparation will give you a competitive edge in the Quantitative section.

**Conclusion: **In this second part of our unit digit cycle series, we explored the concept of the unit digit cycle and its significance in GRE mathematics. Understanding the cyclical nature of unit digits allows you to predict the unit digits of numbers raised to different powers, simplifying calculations and problem-solving. In the upcoming parts, we will dive deeper into formulas, tricks, techniques, general tips, and time management strategies and provide examples of different types of questions involving unit digits.

Stay tuned for Part 3: Formulas, where we will introduce specific formulas related to unit digits and demonstrate their application in solving GRE math problems. Building on this knowledge will further enhance your proficiency in tackling unit digit-related questions on the GRE.

At Max Score Academy, we are committed to your success on the GRE, and we look forward to guiding you through the intricacies of unit digits.

## Part 3: Formulas – Unlocking the Power of Unit Digits

Welcome back to Max Score Academy’s GRE blog series on unit digits. In this third part, we will introduce specific formulas related to unit digits and demonstrate their application in solving GRE math problems. By mastering these formulas, you will have valuable tools at your disposal for tackling various types of questions involving unit digits.

**Formula 1: **Unit Digit of a Sum When adding two or more numbers, the unit digit of the sum is determined solely by the unit digits of the individual numbers being added.

**Example: **Consider the sum 378 + 549. To find the unit digit of the sum, we only need to look at the unit digits of the two numbers: 8 and 9. Adding them together, we get 17. Therefore, the unit digit of the sum is 7.

**Formula 2: **Unit Digit of a Product When multiplying two or more numbers, the unit digit of the product is determined solely by the unit digits of the individual numbers being multiplied.

**Example: **Let’s multiply 74 by 86. To find the unit digit of the product, we only need to look at the unit digits of the two numbers: 4 and 6. Multiplying them together, we get 24. Therefore, the unit digit of the product is 4.

**Formula 3: **Unit Digit of Exponentiation When raising a number to an exponent, the unit digit of the result follows a pattern based on the unit digit of the base number.

**Example: **Suppose we want to find the unit digit of 3^{5}. By referring to the unit digit cycle for 3 (3, 9, 7, 1), we can determine that the unit digit of 3^{5} is the same as the unit digit of 3^1, which is 3.

**Formula 4: **Unit Digit of Repeated Exponentiation When a number is repeatedly raised to the same exponent, the unit digit of the result follows a pattern based on the unit digit of the base number.

Example: Let’s consider 7 raised to different powers: 7^{2}, 7^{4}, 7^{6}, and so on. By referring to the unit digit cycle for 7 (7, 9, 3, 1), we can determine that the unit digits follow the pattern 9, 1, 9, 1. Therefore, 7^2 has a unit digit of 9, 7^{4} has a unit digit of 1, 7^{6} has a unit digit of 9, and so on.

**Utilizing Formulas in Problem Solving:** These formulas are essential tools for simplifying calculations and solving unit digit-related questions efficiently. By applying the appropriate formula based on the given problem, you can quickly determine the unit digit of a sum, product, or exponentiation, eliminating answer choices and saving valuable time.

**Conclusion: **In this third part of our series on unit digits, we have introduced specific formulas that enable you to determine the unit digit of a sum, product, or exponentiation. By leveraging these formulas in your problem-solving approach, you can simplify calculations and make educated choices in selecting the correct unit digit. In the upcoming parts, we will explore tricks, techniques, general tips, and time management strategies and provide examples of different types of questions involving unit digits.

Stay tuned for Part 4: Tricks & Techniques, where we will share valuable tricks and techniques for solving unit digit-related questions efficiently. Mastering these strategies will further enhance your ability to excel in the GRE Quantitative section.

At Max Score Academy, we are dedicated to equipping you with the skills and knowledge needed to succeed in the GRE. Let’s continue to Part 4 of our series:

**Part 4: Tricks & Techniques – Streamlining Unit Digit Problem Solving**

Welcome back to Max Score Academy’s GRE blog series on unit digits. In this fourth part, we will share valuable tricks and techniques that will streamline your problem-solving process when dealing with unit digit-related questions. By incorporating these strategies into your approach, you will gain confidence and efficiency in tackling various types of problems involving unit digits.

**Trick 1: **The Power of Modulo Arithmetic Modulo arithmetic is a powerful technique that simplifies calculations involving remainders. When dealing with unit digits, applying modulo arithmetic can help identify patterns and determine the unit digit of a number.

**Example: **Suppose we want to find the remainder when 345 is divided by 10. We can apply modulo arithmetic by dividing 345 by 10 and considering the remainder: 345 mod 10 = 5. Therefore, the unit digit of 345 is 5.

**Trick 2: **Leveraging Zeroes at the End When multiplying numbers, trailing zeroes at the end of one or more factors will result in a product with the same number of trailing zeroes. This knowledge can be used to quickly determine the unit digit of the product.

**Example: **Let’s multiply 52 by 20. Instead of performing the entire multiplication, we can focus on the unit digit of each number: 2 and 0. Since there is a trailing zero in 20, the unit digit of the product will be zero. Therefore, the unit digit of 52 multiplied by 20 is 0.

**Trick 3: **Recognizing Patterns in Exponents Exponents often exhibit patterns when raised to consecutive powers. Identifying these patterns allows you to determine the unit digit of a number without calculating each individual exponentiation.

Example: Consider 2 raised to different powers: 2^1, 2^2, 2^3, and so on. By observing the pattern in the unit digit cycle of 2 (2, 4, 8, 6), we can deduce that the unit digits repeat every 4 powers. Therefore, the unit digit of 2^21 is the same as the unit digit of 2^1, which is 2.

**Technique: **Backsolving and Elimination When faced with multiple-choice questions involving unit digits, backsolving and elimination techniques can be effective. Start by plugging in answer choices and evaluating their unit digits using the given information or applying relevant formulas. Eliminate answer choices that do not match the expected unit digit, narrowing down your options and increasing the likelihood of selecting the correct answer.

**Conclusion: **In this fourth part of our unit digits series, we explored valuable tricks and techniques for streamlining problem-solving. By utilizing modulo arithmetic, leveraging trailing zeroes, recognizing patterns in exponents, and employing backsolving and elimination, you can approach unit digit-related questions efficiently and accurately. In the upcoming parts, we will delve into general tips, and time management strategies and provide examples of different types of questions involving unit digits.

Stay tuned for Part 5: General Tips and Time Management Tips, where we will share expert advice on maximizing your efficiency and time management when dealing with unit digits on the GRE. At Max Score Academy, we are committed to your success on the GRE Quantitative section and providing you with the necessary tools to excel.

## General Tips and Time Management Tips – Mastering Unit Digits on the GRE

Welcome back to Max Score Academy’s GRE blog series on unit digits. In this fifth part, we will share general tips and time management strategies to help you master unit digits and optimize your performance on the GRE Quantitative section. Incorporating these tips into your preparation and exam approach’ll enhance your efficiency and accuracy when dealing with unit digit-related questions.

General Tips for Unit Digit Problems:

- Review the Unit Digit Cycle: Familiarize yourself with the unit digit cycle for numbers 0 to 9. Understanding the patterns and repetitions in the unit digit cycle will help you quickly determine unit digits in various calculations.
- Practice Mental Math: Strengthen your mental math skills, as unit digit problems often require quick calculations. Practice techniques like multiplication shortcuts, addition/subtraction tricks, and estimation to solve problems swiftly.
- Simplify Complex Expressions: When encountering complex expressions involving unit digits, simplify them by focusing on the unit digits only. This approach reduces the complexity of calculations and eliminates unnecessary steps.
- Be Mindful of Zeroes: Zeroes at the end of a number affect its unit digit. Please pay attention to trailing zeroes when multiplying or performing operations with numbers, as they can determine the unit digit of the result.

**Time Management Tips for Unit Digit Problems:**

- Read the Question Carefully: Understand the specific requirements of the question related to unit digits. Identify whether you need to find the unit digit of a sum, product, exponentiation, or another type of operation.
- Apply Relevant Formulas and Techniques: Based on the question type, apply the appropriate formulas, tricks, or techniques discussed in previous parts of this series. Utilizing the most relevant method saves time and increases accuracy.
- Use Backsolving and Elimination: For multiple-choice questions, consider using the backsolving and elimination technique. Start with the answer choices and substitute them into the problem, checking if the unit digit matches the given information or expected result. This approach narrows down options and improves your chances of selecting the correct answer.
- Prioritize Efficiency: Maintain a steady pace while solving unit digit problems. If you encounter a question that seems time-consuming or difficult, consider marking it for review and returning to it later. Prioritize questions that you can solve quickly and accurately to maximize your score within the allocated time.

**Conclusion:**

In this fifth part of our series on unit digits, we have shared general tips and time management strategies to help you excel in unit digit-related problems on the GRE. You can confidently approach these questions by reviewing the unit digit cycle, practicing mental math, simplifying expressions, and applying relevant formulas. Additionally, implementing effective time management techniques and being mindful of question requirements will further enhance your performance.

Stay tuned for Part 6: Types of Questions with Examples, where we will provide you with various types of questions related to unit digits and walk you through their solutions. At Max Score Academy, we are dedicated to your success in the GRE Quantitative section, and we aim to equip you with the knowledge and strategies needed to conquer unit-digit problems.

**Part 6: Types of Questions with Examples – Applying Unit Digits in Practice**

Welcome back to Max Score Academy’s GRE blog series on unit digits. In this final part, we will explore different types of questions involving unit digits and provide examples to demonstrate how to apply the concepts and techniques we’ve covered throughout the series. Practicing these examples will strengthen your understanding and gain confidence in solving unit digit-related problems on the GRE.

Type 1: Finding the Unit Digit of an Expression

**Example 1:** Find the unit digit of (73 + 89) × 42. Solution: We start by finding the sum of 73 and 89, which is 162. The unit digit of the sum is 2. Next, we multiply the sum by 42. Since the unit digit of 42 is 2, multiplying 2 by 2 gives us a unit digit of 4. Therefore, the unit digit of the expression is 4.

**Type 2: **Determining the Unit Digit of a Product

**Example 2: **Find the unit digit of 67 × 25 × 81. Solution: To determine the unit digit of the product, we focus on the unit digits of each number. The unit digit of 67 is 7, the unit digit of 25 is 5, and the unit digit of 81 is 1. We get 7 × 5 × 1 = 35 by multiplying the unit digits. The unit digit of the product is 5.

**Type 3: **Evaluating the Unit Digit of an Exponentiation

**Example 3:** Determine the unit digit of 8^{37}. Solution: We refer to the unit digit cycle of 8, which repeats every 4 powers (8, 4, 2, 6). Since 37 is not a multiple of 4, we know that the unit digit of 8^37 is the same as the unit digit of 8^1, which is 8.

**Type 4:** Identifying the Unit Digit of a Recursive Sequence

**Example 4: **Find the unit digit of the sequence 3, 3^{3}, 3^{(3^3),} … Solution: We observe that each term in the sequence is obtained by raising 3 to a power that is the previous term in the sequence. We start by finding the unit digit of 3^{3}, which is 7. Then, we find the unit digit of 3^{7}, which is 3. Continuing this pattern, we determine that the unit digit of 3^{(3^3)} is 3. Therefore, the unit digit of the given sequence is 3.

By practicing these examples and similar questions, you will become more proficient in applying unit digits to solve GRE math problems. Remember to utilize the formulas, tricks, techniques, general tips, and time management strategies discussed throughout the series to optimize your performance.

**Conclusion:**

In this final part of our series on unit digits, we explored various types of questions and provided examples to demonstrate how to apply unit digit concepts in practice. By understanding how to find the unit digit of an expression, determine the unit digit of a product, evaluate the unit digit of an exponentiation, and identify the unit digit of a recursive sequence, you will be well-prepared to handle unit digit-related questions on the GRE.

Max Score Academy is dedicated to your success on the GRE Quantitative section, and we hope that this series has equipped you with the necessary tools to master unit digits. Best of luck in your GRE preparation and future endeavors!