Units Digit Calculation: Solving For X And Y

Hey guys! Ever found yourself staring at a math problem that seems intimidating at first glance? Well, today we're going to break down one such problem together, making sure we not only get to the solution but also understand the why behind it. Let's dive into a cool problem involving finding the units digits of some pretty large numbers. We've got $x = 7^{2025}$ and $y = 3^{2026}$, and our mission, should we choose to accept it, is to figure out the units digits of these numbers and their combinations. Sounds like fun, right? Let's jump in!

Unpacking the Problem

So, the question we're tackling today involves finding the units digits of expressions involving exponents. Specifically, we're given $x = 7^{2025}$ and $y = 3^{2026}$. Our goal is to determine the units digit of each of these numbers, as well as the units digit of their sum, difference, and product. This means we need to figure out which of the following statements are true:

• The units digit of $x$ is 7
• The units digit of $y$ is 3
• The units digit of $x + y$ is 6
• The units digit of $x - y$ is 4
• The units digit of $xy$ is 3

Now, before we get lost in the vastness of these exponents, there's a neat trick we can use. Instead of calculating these massive powers directly, we can focus on the cyclical nature of units digits. Think of it like this: when you multiply a number by itself repeatedly, the units digit often follows a pattern. Let's explore these patterns for 7 and 3, which are the base numbers in our expressions. By understanding these cycles, we can simplify our calculations and make this problem a whole lot easier. So, buckle up, and let's get started on unraveling these digits!

Cracking the Code: Finding the Units Digit of $x = 7^{2025}$

Okay, let's start with $x = 7^{2025}$. The key here is to look for a pattern in the units digits of the powers of 7. We don't need to calculate the entire number; we just need to track the last digit. So, let's write out the first few powers of 7 and see what happens:

• $7^1 = 7$
• $7^2 = 49$ (units digit is 9)
• $7^3 = 343$ (units digit is 3)
• $7^4 = 2401$ (units digit is 1)
• $7^5 = 16807$ (units digit is 7)

Notice anything? The units digits repeat in a cycle: 7, 9, 3, 1. This cycle has a length of 4. That means every fourth power of 7 will have a units digit of 1, and then the cycle starts again. This is crucial for solving our problem without actually computing $7^{2025}$.

Now, to find the units digit of $7^{2025}$, we need to figure out where 2025 falls within this cycle. We can do this by dividing the exponent, 2025, by the length of the cycle, which is 4. So, let's do the math: $2025 ext{ ÷ } 4 = 506$ with a remainder of 1. This remainder is super important because it tells us where we are in the cycle. A remainder of 1 means that $7^{2025}$ will have the same units digit as $7^1$. So, what's the units digit of $7^1$? It's 7! Therefore, the units digit of $x = 7^{2025}$ is 7. We've just cracked the code for $x$, guys! Now, let's move on to $y$ and see if we can find its units digit using a similar method.

Decoding the Units Digit of $y = 3^{2026}$

Alright, now let's tackle $y = 3^{2026}$. We'll use the same trick we used for $x$: find the pattern in the units digits of powers of 3. Let's list out the first few:

• $3^1 = 3$
• $3^2 = 9$
• $3^3 = 27$ (units digit is 7)
• $3^4 = 81$ (units digit is 1)
• $3^5 = 243$ (units digit is 3)

See the pattern? The units digits here also form a cycle: 3, 9, 7, 1. Just like with the powers of 7, this cycle has a length of 4. So, every fourth power of 3 will have a units digit of 1, and then the cycle repeats itself. Understanding this cyclical behavior is key to finding the units digit of $3^{2026}$.

To figure out the units digit of $3^{2026}$, we need to see where 2026 falls in this cycle. We'll divide the exponent, 2026, by the cycle length, which is 4. Let's do it: $2026 ext{ ÷ } 4 = 506$ with a remainder of 2. That remainder is our magic number! A remainder of 2 means that $3^{2026}$ will have the same units digit as $3^2$. And what's the units digit of $3^2$? It's 9! So, the units digit of $y = 3^{2026}$ is 9. Fantastic! We've decoded the units digit for both $x$ and $y$. Now, the fun really begins as we start combining these results to figure out the units digits of $x+y$, $x-y$, and $xy$. Let's keep going!

Putting It All Together: Finding the Units Digits of $x+y$, $x-y$, and $xy$

Okay, now that we know the units digit of $x$ is 7 and the units digit of $y$ is 9, we can tackle the remaining parts of the problem. Remember, we need to find the units digits of $x+y$, $x-y$, and $xy$. This is where our previous work really pays off, making these calculations surprisingly straightforward.

Units Digit of $x + y$

To find the units digit of $x + y$, we simply add the units digits of $x$ and $y$. So, we have $7 + 9 = 16$. The units digit of 16 is 6. Therefore, the units digit of $x + y$ is 6. See? Not too shabby! This step shows how understanding the cyclical nature of units digits can make seemingly complex problems much simpler.

Units Digit of $x - y$

Next, let's find the units digit of $x - y$. Here, we subtract the units digit of $y$ from the units digit of $x$. So, we have $7 - 9$. Now, this gives us a negative number, -2. To find the units digit in this case, we need to add 10 to the result (since we're working in base 10). So, $-2 + 10 = 8$. Therefore, the units digit of $x - y$ is 8. It's important to remember this trick when subtracting units digits and ending up with a negative number.

Units Digit of $xy$

Finally, let's find the units digit of $xy$. We multiply the units digits of $x$ and $y$. So, we have $7 imes 9 = 63$. The units digit of 63 is 3. Therefore, the units digit of $xy$ is 3. This calculation wraps up our individual units digit findings.

Now that we've found the units digits of $x + y$, $x - y$, and $xy$, we can revisit the original statements and see which ones hold true. Let's recap our findings and then evaluate the given options. Are you ready to see how it all comes together? Let’s do it!

Wrapping It Up: Identifying the Correct Statements

Alright, let's recap what we've found. We've determined the following:

• The units digit of $x = 7^{2025}$ is 7.
• The units digit of $y = 3^{2026}$ is 9.
• The units digit of $x + y$ is 6.
• The units digit of $x - y$ is 8.
• The units digit of $xy$ is 3.

Now, let's revisit the statements and see which ones are correct:

• The units digit of $x$ is 7: This is correct, as we found.
• The units digit of $y$ is 3: This is incorrect; the units digit of $y$ is 9.
• The units digit of $x + y$ is 6: This is correct, as we calculated.
• The units digit of $x - y$ is 4: This is incorrect; the units digit of $x - y$ is 8.
• The units digit of $xy$ is 3: This is correct, based on our calculations.

So, the correct statements are:

• The units digit of $x$ is 7.
• The units digit of $x + y$ is 6.
• The units digit of $xy$ is 3.

We did it! We took a seemingly complex problem and broke it down into manageable steps by focusing on the cyclical nature of units digits. This approach not only helped us find the solution but also deepened our understanding of number patterns. Remember, guys, in math, there's often more than one way to tackle a problem, and looking for patterns can be a powerful tool in your arsenal. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!