Finding the rightmost digit of a power can seem like a daunting task, but it becomes much simpler once you understand the underlying patterns and techniques. Let’s break it down step by step.
Understanding the Basics
When we talk about finding the rightmost digit of a power, we’re essentially looking for the last digit of a number like $2^{10}$ or $7^{5}$. The rightmost digit of a number is the same as the remainder when the number is divided by 10.
Example 1: $2^{10}$
Instead of calculating $2^{10}$ directly, which equals 1024, we can focus on the last digit of each power of 2:
- $2^1 = 2$
- $2^2 = 4$
- $2^3 = 8$
- $2^4 = 16$ (last digit is 6)
- $2^5 = 32$ (last digit is 2)
Notice that the last digits start repeating every four powers: 2, 4, 8, 6. This repeating pattern is called a cycle. For $2^{10}$, the last digit is the same as $2^2$ because 10 mod 4 equals 2. So, the rightmost digit of $2^{10}$ is 4.
Steps to Find the Rightmost Digit
Identify the Base and Exponent
Let’s say we want to find the rightmost digit of $a^b$
Determine the Cycle Length
Calculate the first few powers of the base (a) until you notice a repeating pattern in the last digits. This pattern is the cycle.
Use Modular Arithmetic
Find the position of the exponent (b) within the cycle using modular arithmetic. Specifically, calculate $b bmod text{cycle length}$. This gives you the position within the cycle that corresponds to the last digit of $a^b$
Detailed Examples
Example 2: $7^5$
Identify the Base and Exponent: Here, $a = 7$ and $b = 5$
Determine the Cycle Length:
- $7^1 = 7$
- $7^2 = 49$ (last digit is 9)
- $7^3 = 343$ (last digit is 3)
- $7^4 = 2401$ (last digit is 1)
The cycle is 7, 9, 3, 1, and it repeats every 4 powers.
Use Modular Arithmetic: Calculate $5 bmod 4 = 1$. This means $7^5$ has the same last digit as $7^1$, which is 7.
Example 3: $3^{14}$
Identify the Base and Exponent: Here, $a = 3$ and $b = 14$
Determine the Cycle Length:
- $3^1 = 3$
- $3^2 = 9$
- $3^3 = 27$ (last digit is 7)
- $3^4 = 81$ (last digit is 1)
The cycle is 3, 9, 7, 1, and it repeats every 4 powers.
Use Modular Arithmetic: Calculate $14 bmod 4 = 2$. This means $3^{14}$ has the same last digit as $3^2$, which is 9.
Special Cases
Powers of 0 and 1
- The rightmost digit of $0^n$ (where $n > 0$) is always 0.
- The rightmost digit of $1^n$ is always 1.
Powers of 5 and 6
- The rightmost digit of $5^n$ is always 5.
- The rightmost digit of $6^n$ is always 6.
These cases are straightforward because they don’t form cycles and their last digits remain constant.
Practice Problems
- Find the rightmost digit of $9^{12}$
- Find the rightmost digit of $8^{15}$
- Find the rightmost digit of $4^{23}$
Solutions
Problem 1: $9^{12}$
Identify the Base and Exponent: Here, $a = 9$ and $b = 12$
Determine the Cycle Length:
- $9^1 = 9$
- $9^2 = 81$ (last digit is 1)
The cycle is 9, 1, and it repeats every 2 powers.
Use Modular Arithmetic: Calculate $12 bmod 2 = 0$. This means $9^{12}$ has the same last digit as $9^0$, which is 1.
Problem 2: $8^{15}$
Identify the Base and Exponent: Here, $a = 8$ and $b = 15$
Determine the Cycle Length:
- $8^1 = 8$
- $8^2 = 64$ (last digit is 4)
- $8^3 = 512$ (last digit is 2)
- $8^4 = 4096$ (last digit is 6)
The cycle is 8, 4, 2, 6, and it repeats every 4 powers.
Use Modular Arithmetic: Calculate $15 bmod 4 = 3$. This means $8^{15}$ has the same last digit as $8^3$, which is 2.
Problem 3: $4^{23}$
Identify the Base and Exponent: Here, $a = 4$ and $b = 23$
Determine the Cycle Length:
- $4^1 = 4$
- $4^2 = 16$ (last digit is 6)
The cycle is 4, 6, and it repeats every 2 powers.
Use Modular Arithmetic: Calculate $23 bmod 2 = 1$. This means $4^{23}$ has the same last digit as $4^1$, which is 4.
Conclusion
Finding the rightmost digit of a power is all about identifying patterns and using modular arithmetic. By breaking down the problem into smaller steps, you can easily determine the last digit without performing large calculations. This method is not only efficient but also helps you understand the cyclical nature of numbers.
Happy calculating!