GRE Unit Digit Number Properties

GRE Unit Digit Number Properties
MKS Education’s Comprehensive Guide to GRE

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Unraveling the Unit Digit Number Property – Introduction and Fundamentals

Hello, aspiring GRE candidates! Welcome to MKS Education’s GRE blog series. Today, we embark on an exciting journey into the world of number properties, focusing on the Unit Digit Number Property.

The unit digit number property is a fundamental mathematical concept frequently tested in competitive exams like the GRE. Mastering this concept not only simplifies your quantitative section but also enhances your overall mathematical skills.

So, what is a unit digit? It’s the digit in the ones place of a number. For example, the unit digit of 12345 is 5, and for 6789, it’s 9. While this may seem simple, the unit digit holds fascinating patterns and applications that we’ll explore in this five-part series.

Stay tuned as we dive into the basics, uncover patterns, apply the concept to GRE problems, and even tackle advanced applications. By the end of this series, you’ll confidently handle any GRE question involving unit digits.

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Unraveling the Unit Digit Number Property – Patterns and Cycles

One of the most intriguing aspects of unit digits is their cyclical patterns when numbers are raised to powers. This means that unit digits repeat in a specific sequence after a certain interval.

For example, consider the number 2. Let’s examine the unit digits of its successive powers:

• 2¹ = 2
• 2² = 4
• 2³ = 8
• 2⁴ = 6
• 2⁵ = 2

Here, the unit digits 2, 4, 8, 6 repeat every 4 powers. This cycle continues indefinitely for higher powers of 2.

Recognizing these patterns is crucial for the GRE, as questions often involve finding the unit digit of large exponents. Instead of calculating the entire number, you can use these cycles to quickly determine the unit digit, saving time and improving accuracy.

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Unraveling the Unit Digit Number Property – Application in GRE Problems

A common GRE question type is: “What is the unit digit of a large power?” Let’s solve an example using the cyclical patterns we’ve learned.

Question: What is the unit digit of 7⁷⁵?

Solution:

1. The unit digit of 7 follows a cycle of 7, 9, 3, 1 every 4 powers.
2. Divide the exponent (75) by the cycle length (4):
   • 75 ÷ 4 = 18 with a remainder of 3.
3. The remainder tells us the position in the cycle:
   • Remainder 3 corresponds to the third digit in the cycle, which is 3.

Thus, the unit digit of 7⁷⁵ is 3.

This method is efficient and strategic, helping you tackle similar problems quickly during the exam.

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Unraveling the Unit Digit Number Property – Advanced Applications and Interactions

The unit digit property also interacts with other mathematical concepts, such as divisibility and remainders.

1. Divisibility Rules:
   • A number is divisible by 2 if its unit digit is even (0, 2, 4, 6, 8).
   • A number is divisible by 5 if its unit digit is 0 or 5.
   • A number is divisible by 10 if its unit digit is 0.
2. Remainders:
   • The unit digit can help determine remainders when dividing by certain numbers. For example, a number ending in 1, 5, 6, or 0 has predictable remainders when divided by other numbers.
3. Sums and Products:
   • The unit digit of a sum or product depends only on the unit digits of the numbers involved. For example, the unit digit of (26 + 34) is the same as the unit digit of (6 + 4), which is 0.

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

Now that we’ve covered the fundamentals, let’s test your understanding with practice questions and a challenge problem.

Practice Question 1:

What is the unit digit of 2⁵⁷ + 3⁶⁶?

Solution:

1. For 2⁵⁷:
   • The cycle for 2 is 2, 4, 8, 6.
   • 57 ÷ 4 leaves a remainder of 1, so the unit digit is 2.
2. For 3⁶⁶:
   • The cycle for 3 is 3, 9, 7, 1.
   • 66 ÷ 4 leaves a remainder of 2, so the unit digit is 9.
3. The unit digit of the sum is 2 + 9 = 11, which has a unit digit of 1.

Answer: 1

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Practice Question 2:

If n is a positive integer, what is the unit digit of 7^(4n+3)?

Solution:

1. The cycle for 7 is 7, 9, 3, 1.
2. The exponent (4n + 3) ensures the unit digit is always the third digit in the cycle, which is 3.

Answer: 3

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Challenge Problem:

What is the unit digit of (5¹⁰⁰ – 3¹⁰⁰)?

Solution:

1. The unit digit of any power of 5 is always 5.
2. For 3¹⁰⁰:
   • The cycle for 3 is 3, 9, 7, 1.
   • 100 ÷ 4 leaves no remainder, so the unit digit is 1.
3. The unit digit of the expression is 5 – 1 = 4.

Answer: 4

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Conclusion

We hope this series has deepened your understanding of the Unit Digit Number Property and its applications. Practice is key to mastering this concept, so keep solving problems and honing your skills.

At MKS Education, we’re committed to providing you with the tools and knowledge to excel on the GRE. Stay tuned for more insightful series on essential GRE topics. Here’s to your success on the GRE and beyond!

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Unraveling the Unit Digit Number Property – Practice Questions with Answer Key

To solidify your understanding of the unit digit number property, here are some additional practice questions along with detailed solutions. These problems are designed to help you apply the concepts of cyclical patterns, divisibility, and advanced applications.

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Practice Question 1:

What is the unit digit of 8⁴⁵?

Solution:

1. The unit digit of 8 follows a cycle of 8, 4, 2, 6 every 4 powers.
2. Divide the exponent (45) by the cycle length (4):
   • 45 ÷ 4 = 11 with a remainder of 1.
3. The remainder tells us the position in the cycle:
   • Remainder 1 corresponds to the first digit in the cycle, which is 8.

Answer: 8

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Practice Question 2:

What is the unit digit of 9²⁷?

Solution:

1. The unit digit of 9 follows a cycle of 9, 1 every 2 powers.
2. Divide the exponent (27) by the cycle length (2):
   • 27 ÷ 2 = 13 with a remainder of 1.
3. The remainder tells us the position in the cycle:
   • Remainder 1 corresponds to the first digit in the cycle, which is 9.

Answer: 9

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Practice Question 3:

What is the unit digit of 4⁶³ + 6⁴²?

Solution:

1. For 4⁶³:
   • The cycle for 4 is 4, 6 every 2 powers.
   • 63 ÷ 2 leaves a remainder of 1, so the unit digit is 4.
2. For 6⁴²:
   • The unit digit of any power of 6 is always 6.
3. The unit digit of the sum is 4 + 6 = 10, which has a unit digit of 0.

Answer: 0

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Practice Question 4:

What is the unit digit of 12³⁴ × 13³⁵?

Solution:

1. Focus on the unit digits of the base numbers:
   • The unit digit of 12 is 2, and the unit digit of 13 is 3.
2. For 2³⁴:
   • The cycle for 2 is 2, 4, 8, 6 every 4 powers.
   • 34 ÷ 4 leaves a remainder of 2, so the unit digit is 4.
3. For 3³⁵:
   • The cycle for 3 is 3, 9, 7, 1 every 4 powers.
   • 35 ÷ 4 leaves a remainder of 3, so the unit digit is 7.
4. The unit digit of the product is 4 × 7 = 28, which has a unit digit of 8.

Answer: 8

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Practice Question 5:

What is the unit digit of (7¹² + 5¹⁵) × 9²⁰?

Solution:

1. For 7¹²:
   • The cycle for 7 is 7, 9, 3, 1 every 4 powers.
   • 12 ÷ 4 leaves no remainder, so the unit digit is 1.
2. For 5¹⁵:
   • The unit digit of any power of 5 is always 5.
3. For 9²⁰:
   • The cycle for 9 is 9, 1 every 2 powers.
   • 20 ÷ 2 leaves no remainder, so the unit digit is 1.
4. The unit digit of the sum inside the parentheses is 1 + 5 = 6.
5. The unit digit of the product is 6 × 1 = 6.

Answer: 6

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Challenge Problem:

What is the unit digit of (2¹⁰⁰ + 3¹⁰⁰ + 4¹⁰⁰ + 5¹⁰⁰)?

Solution:

1. For 2¹⁰⁰:
   • The cycle for 2 is 2, 4, 8, 6 every 4 powers.
   • 100 ÷ 4 leaves no remainder, so the unit digit is 6.
2. For 3¹⁰⁰:
   • The cycle for 3 is 3, 9, 7, 1 every 4 powers.
   • 100 ÷ 4 leaves no remainder, so the unit digit is 1.
3. For 4¹⁰⁰:
   • The cycle for 4 is 4, 6 every 2 powers.
   • 100 ÷ 2 leaves no remainder, so the unit digit is 6.
4. For 5¹⁰⁰:
   • The unit digit of any power of 5 is always 5.
5. The unit digit of the sum is 6 + 1 + 6 + 5 = 18, which has a unit digit of 8.

Answer: 8

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Answer Key Summary

1. 8⁴⁵: 8
2. 9²⁷: 9
3. 4⁶³ + 6⁴²: 0
4. 12³⁴ × 13³⁵: 8
5. (7¹² + 5¹⁵) × 9²⁰: 6
6. (2¹⁰⁰ + 3¹⁰⁰ + 4¹⁰⁰ + 5¹⁰⁰): 8

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Final Thoughts

Practicing these problems will help you master the unit digit number property and its applications. Remember to identify cycles, use remainders, and apply divisibility rules to solve problems efficiently.

At MKS Education, we’re here to support your GRE preparation journey. Keep practicing, and stay tuned for more insightful guides! Let us know in the comments if you have any questions or need further clarification. Good luck! 🚀