Finding The Last Digit: A Mathematical Adventure

by TextBrain Team 49 views

Hey math enthusiasts! Today, we're diving into a super fun topic: figuring out the last digit of some seriously big numbers. Don't worry, it's not as scary as it sounds. We'll be looking at powers of numbers like 2, 3, 5, 6, 7, 8, 9, and 4, all raised to some hefty exponents. Our mission? To crack the code and determine what the final digit of each number will be. It's like a mathematical detective story, and we're the investigators! Let's break down how to solve this math puzzle, step by step, so you can become a last-digit master.

Powers of 2: Unveiling the Pattern

Alright, let's kick things off with powers of 2. Our first number is 2 to the power of 2017 (2^2017). To find the last digit, we don't need to calculate the entire number (which would be enormous!). Instead, we can spot a repeating pattern. Let's look at the first few powers of 2 and their last digits:

  • 2^1 = 2 (Last digit: 2)
  • 2^2 = 4 (Last digit: 4)
  • 2^3 = 8 (Last digit: 8)
  • 2^4 = 16 (Last digit: 6)
  • 2^5 = 32 (Last digit: 2)
  • 2^6 = 64 (Last digit: 4)

See the pattern? The last digits repeat in a cycle: 2, 4, 8, 6. This cycle has a length of 4. To find the last digit of 2^2017, we need to figure out where 2017 falls within this cycle. We do this by dividing the exponent (2017) by the cycle length (4).

2017 divided by 4 gives us 504 with a remainder of 1. This remainder is super important! It tells us where we land in the cycle. A remainder of 1 means the last digit of 2^2017 is the first number in our cycle, which is 2. So, the last digit of 2^2017 is 2. We've solved the first mystery!

Let's recap: to find the last digit of a number raised to a power, look for a repeating pattern in the last digits of the powers of that number. Divide the exponent by the cycle length. The remainder tells you which number in the cycle is the last digit of your original number. Pretty cool, right?

Powers of 3: Another Pattern Emerges

Next up, we're tackling powers of 3. We're looking at 3 to the power of 2017 (3^2017). Time to unveil another pattern. Let's list out the first few powers of 3 and their last digits:

  • 3^1 = 3 (Last digit: 3)
  • 3^2 = 9 (Last digit: 9)
  • 3^3 = 27 (Last digit: 7)
  • 3^4 = 81 (Last digit: 1)
  • 3^5 = 243 (Last digit: 3)
  • 3^6 = 729 (Last digit: 9)

And there it is! The last digits form a repeating cycle: 3, 9, 7, 1. This cycle also has a length of 4. Just like before, we'll divide the exponent (2017) by the cycle length (4).

Again, 2017 divided by 4 gives us 504 with a remainder of 1. A remainder of 1 means we start at the beginning of our cycle. Therefore, the last digit of 3^2017 is 3. Another case closed!

Remember, the key is to find that repeating pattern. Once you've got the cycle, it's smooth sailing.

Powers of 5 and 6: Easy Wins

Now, let's take a breather because the powers of 5 and 6 are a walk in the park. Seriously, you'll love this.

  • Powers of 5: Any power of 5 will always end in 5. It's a simple rule! 5^2018 will end in 5.
  • Powers of 6: Similarly, any power of 6 will always end in 6. 6^2019 will end in 6.

See? Told you it was easy. Some problems are just meant to be straightforward.

Powers of 7: Back to the Cycle

Alright, let's get back to business and explore powers of 7. We're working with 7 to the power of 2020 (7^2020). Time to uncover another pattern:

  • 7^1 = 7 (Last digit: 7)
  • 7^2 = 49 (Last digit: 9)
  • 7^3 = 343 (Last digit: 3)
  • 7^4 = 2401 (Last digit: 1)
  • 7^5 = 16807 (Last digit: 7)

The cycle is 7, 9, 3, 1, which also has a length of 4. To find the last digit of 7^2020, we divide the exponent (2020) by the cycle length (4).

2020 divided by 4 is 505 with a remainder of 0. A remainder of 0 means the last digit is the last number in the cycle, which is 1. Therefore, the last digit of 7^2020 is 1.

Powers of 8: One More Cycle to Conquer

Let's keep the momentum going with powers of 8. We're looking at 8 to the power of 2021 (8^2021). Here's the pattern:

  • 8^1 = 8 (Last digit: 8)
  • 8^2 = 64 (Last digit: 4)
  • 8^3 = 512 (Last digit: 2)
  • 8^4 = 4096 (Last digit: 6)
  • 8^5 = 32768 (Last digit: 8)

The cycle here is 8, 4, 2, 6. This cycle also has a length of 4. We need to divide the exponent (2021) by the cycle length (4).

2021 divided by 4 is 505 with a remainder of 1. A remainder of 1 means the last digit is the first number in the cycle, which is 8. Thus, the last digit of 8^2021 is 8.

Powers of 9: Almost There!

Almost to the finish line! Let's look at powers of 9. We have 9 to the power of 2022 (9^2022). Let's uncover the pattern:

  • 9^1 = 9 (Last digit: 9)
  • 9^2 = 81 (Last digit: 1)
  • 9^3 = 729 (Last digit: 9)

See the cycle? It's a short one: 9, 1. This cycle has a length of 2. We divide the exponent (2022) by the cycle length (2).

2022 divided by 2 is 1011 with a remainder of 0. A remainder of 0 means the last digit is the last number in the cycle, which is 1. Therefore, the last digit of 9^2022 is 1.

Powers of 4: The Final Challenge

Finally, we are on the last number 4 to the power of 2023 (4^2023). Let's find the repeating pattern:

  • 4^1 = 4 (Last digit: 4)
  • 4^2 = 16 (Last digit: 6)
  • 4^3 = 64 (Last digit: 4)

The cycle is 4, 6. This cycle has a length of 2. We need to divide the exponent (2023) by the cycle length (2).

2023 divided by 2 is 1011 with a remainder of 1. A remainder of 1 means the last digit is the first number in the cycle, which is 4. Thus, the last digit of 4^2023 is 4.

Conclusion: You Did It!

Congratulations, you've successfully determined the last digits of all our numbers! It’s all about finding those patterns and applying the simple division trick. Now, you're a last-digit expert! Keep practicing, and you'll be able to solve these problems with ease. Happy calculating!