Last Digit Of 2^78 * 6^80: A Math Puzzle

Let's dive into a cool math problem: figuring out the last digit of the number you get when you multiply 2 to the power of 78 by 6 to the power of 80. Sounds tricky, right? But don't worry, we'll break it down step by step. This isn't just about crunching numbers; it's about spotting patterns and using some neat mathematical tricks. So, grab your thinking caps, and let's get started!

Understanding the Problem

At its heart, this problem is about number theory and modular arithmetic. Essentially, we're interested in the remainder when $2^{78} \cdot 6^{80}$ is divided by 10. This remainder will give us the last digit. To tackle this, we'll look at the patterns of the last digits of powers of 2 and powers of 6. By understanding these cyclical patterns, we can simplify the exponents and find the last digit without having to compute the entire number. This approach makes the problem much more manageable and highlights the beauty of number patterns in mathematics. So, before we start calculating, let's explore these patterns and how they can help us solve this problem efficiently. Remember, math is not just about getting the right answer, but also about understanding why the answer is right!

Powers of 2: Spotting the Pattern

When we talk about powers of 2, we're looking at numbers like $2^1, 2^2, 2^3$, and so on. What's super interesting is how the last digits of these numbers form a repeating pattern. Let's list the first few powers of 2 and focus on their last digits:

• $2^1 = 2$ (last digit is 2)
• $2^2 = 4$ (last digit is 4)
• $2^3 = 8$ (last digit is 8)
• $2^4 = 16$ (last digit is 6)
• $2^5 = 32$ (last digit is 2)

See that? The last digits repeat in a cycle: 2, 4, 8, 6. This cycle has a length of 4. So, to find the last digit of $2^{78}$, we need to figure out where 78 falls in this cycle. We can do this by dividing 78 by 4: $78 \div 4 = 19$ with a remainder of 2. This means that $2^{78}$ will have the same last digit as $2^2$, which is 4. This is because after every 4 powers, the cycle repeats. Thus, $2^{78}$ completes 19 full cycles and lands on the second number in the cycle. Understanding this pattern is key to solving the problem without having to calculate large powers of 2. This principle applies to many other numbers as well, making it a valuable tool in number theory.

Powers of 6: An Easy Observation

Now, let's shift our focus to powers of 6. These are numbers like $6^1, 6^2, 6^3$, and so on. Here's a fun fact: no matter what power you raise 6 to, the last digit will always be 6. Let's see why:

• $6^1 = 6$
• $6^2 = 36$
• $6^3 = 216$
• $6^4 = 1296$

Notice that the last digit is always 6. This happens because when you multiply any number ending in 6 by 6, the result will always end in 6. Mathematically, we can express this as $(10n + 6) \cdot 6 = 60n + 36 = 10(6n + 3) + 6$, where $n$ is any integer. The result is always a multiple of 10 plus 6, meaning the last digit is always 6. So, $6^{80}$ will have a last digit of 6. This makes our problem a lot easier because we don't need to worry about a repeating cycle. The last digit of any power of 6 is consistently 6. This is a special property of the number 6, making it stand out in number theory. It's always great to find these shortcuts, isn't it?

Multiplying the Last Digits

Okay, so we've figured out that the last digit of $2^{78}$ is 4, and the last digit of $6^{80}$ is 6. Now, to find the last digit of $2^{78} \cdot 6^{80}$, we simply need to multiply these last digits together: $4 \cdot 6 = 24$. The last digit of 24 is 4. Therefore, the last digit of $2^{78} \cdot 6^{80}$ is 4. This step combines our previous findings to reach the final answer. By breaking down the problem into smaller parts and focusing on the last digits, we avoided the need to calculate large numbers. This approach showcases the power of modular arithmetic and pattern recognition in solving mathematical problems. It's a great example of how understanding the properties of numbers can lead to efficient solutions. Remember, in many mathematical problems, looking for patterns and simplifications is key to finding the answer. This method not only gives us the solution but also enhances our understanding of number theory.

The Final Answer

So, after all that, we've cracked it! The last digit of $2^{78} \cdot 6^{80}$ is 4. Wasn't that a fun little journey through the world of numbers? We used pattern recognition and some basic modular arithmetic to solve this problem without needing a calculator for huge numbers. Remember, math isn't always about complex calculations; sometimes it's about spotting the clever tricks that make things easier. Hopefully, this explanation helped you understand how to approach similar problems in the future. Keep exploring, keep questioning, and you'll find that math can be quite fascinating! And who knows, maybe you'll discover some new patterns and shortcuts of your own.