## GRE Unit Digit Number Properties

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## GRE Unit Digit Number Properties

**MKSprep’s Comprehensive Guide to GRE**

**Unraveling the Unit Digit Number Property – Introduction and Fundamentals**

Hello, aspiring GRE candidates! Welcome to MKSprep’s GRE blog series. Today, we kick-start an exciting new chapter of our course, delving deep into the intriguing world of number properties. Our spotlight for this series? The Unit Digit Number Property.

The unit digit number property is a fundamental mathematical concept, frequently encountered in various competitive exams, including the GRE. Understanding this concept eases your journey through the quantitative section and enhances your overall mathematical acumen.

Now, what exactly is the unit digit? The unit digit of a number is the digit in one place. For instance, the unit digit of 12345 is 5, and for 6789, it’s 9. But there’s much more to this concept than first meets the eye, and that’s precisely what we’ll explore in this five-part blog series.

Stay tuned as we unravel the mysteries of the unit digit number property. We’ll start from the basics, walk through the intricacies, apply the concept to real GRE problems, and even explore advanced applications. By the end of this series, you’ll be well-equipped to tackle any GRE question involving unit digits with confidence and accuracy.

**Unraveling the Unit Digit Number Property – Patterns and Cycles**

When we talk about the unit digit number property, one essential feature that emerges is the cyclical pattern of unit digits. What does this mean? It means that unit digits repeat themselves in a certain pattern after a specific interval when a number is raised to increasing powers.

For example, take the number 2. If we look at the unit digit of successive powers of 2, we see a pattern:

- 2
^{1}= 2 - 2
^{2}= 4 - 2
^{3}= 8 - 2
^{4}= 16 - 2
^{5}= 32

Here, the unit digits 2, 4, 8, and 6 repeat themselves in a cycle every four powers. This cycle of four (2, 4, 8, 6) continues indefinitely for all higher powers of 2.

Recognizing these patterns is essential for the GRE, as questions often involve finding the unit digit of a large power, which would be tedious without knowing these cycles. Understanding and applying these patterns can save precious time during the exam and increase your accuracy in answering such problems.

**Unraveling the Unit Digit Number Property – Application in GRE Problems**

Here’s a common GRE problem type: “What is the unit digit of a large power?” We can handle this without breaking a sweat thanks to what we’ve learned. Let’s illustrate with an example:

Question: What is the unit digit of 7^{75}?

From our understanding of the cyclical patterns of unit digits, we know that the unit digit of 7 repeats every four powers, with the cycle being 7, 9, 3, and 1.

Now, to find the unit digit of 7^{75}, we divide the exponent 75 by the length of the cycle, which is 4. The remainder will tell us which number is the unit digit in the cycle.

75 ÷ 4 = 18 remainder 3.

The third number in the cycle (7, 9, 3, 1) is 3. So, the unit digit of 7^{75} is 3.

This method is quick and efficient, saving you time on the exam and increasing your chances of a higher score. Remember, the GRE is as much about strategy as it is about knowledge.

**Unraveling the Unit Digit Number Property – Advanced Applications and Interactions**

Let’s start by considering a problem where we have to find the unit digit of a sum or a product. Remember, the unit digit of a sum or product is simply the unit digit of the sum or product of the individual unit digits. For example, the unit digit of (2^{6} + 3^{4}) is the same as the unit digit of (4 + 1), which is 5.

Now, let’s consider an intersection with another number property: divisibility. The unit digit can determine whether a number is divisible by 2, 5, or 10. If the unit digit is even (0, 2, 4, 6, 8), the number is divisible by 2. If it’s 0 or 5, the number is divisible by 5; if it’s 0, it is divisible by 10.

Another exciting intersection is with the concept of remainders. For instance, finding the remainder when a number with a unit digit of 1, 5, 6, or 0 is divided by any number can be done by only looking at the unit digit.

**Unraveling the Unit Digit Number Property – Practice Questions and Challenge Problems**

Now that we’ve covered the basics, patterns, applications, and advanced concepts, it’s time to test your knowledge with practice questions and challenging problems.

**Practice Question 1:** What is the unit digit of 2^{57} + 3^{66}?

Solution: The unit digit of 2^{57} is found by dividing 57 by 4 (the length of the cycle for 2), which leaves a remainder of 1. So, the unit digit is the first number in the cycle (2). Similarly, for 3^66, the cycle length is 4, and 66 divided by 4 leaves the remainder 2, so the unit digit is 4, the last number in the cycle (2). The unit digit of the sum is, therefore, 2 +4 = 6.

**Practice Question 2:** If n is a positive integer, what is the unit digit of 7^{(4n+3)?}

Solution: The cycle for 7 is 7, 9, 3, 1. Since the exponent is of form 4n+3, the unit digit is always the third digit in the cycle, which is 3.

**Challenge Problem:** What is the unit digit of (5^{100} – 3^{100})?

Solution: The unit digit of any power of 5 is always 5. For 3^{100}, since 100 divided by 4 leaves no remainder, the unit digit is the last number in the cycle for 3, which is 1. So, the unit digit of the expression is 5 – 1 = 4.

We hope these examples and solutions have further solidified your understanding of the Unit Digit Number Property. Practice is key to mastering any concept, so keep honing your skills.

At MKSprep, we aim to provide you with all the tools you need to succeed in your GRE journey. Stay tuned for more exciting and insightful series on other essential GRE topics. Here’s to your success on the GRE and beyond!