Divisibility By 9 A Detailed Calculation And Analysis
Hey guys! Today, we're diving into a fun mathematical problem: determining if the results of some pretty hefty calculations are divisible by 9. We've got three expressions to crack, and we’ll go through each one step-by-step. So, grab your calculators (or your mental math hats) and let's get started!
Expression A Calculation and Divisibility Analysis
Let's kick things off with expression A: . This looks like a beast, but don't worry, we'll tame it. Our main goal here is to figure out if the final result is divisible by 9. This involves several arithmetic operations, and we need to be precise to get it right. Divisibility rules are super handy, and for 9, it’s all about whether the sum of the digits is divisible by 9. Remember, meticulous calculation is key in problems like these.
First, we tackle the multiplication and division within the parentheses. We start by multiplying 528 by 6996. This gives us an intermediate result, which we then divide by 22. It’s crucial to keep track of these intermediate values because any error here will throw off the entire calculation. Think of it like building a house; if the foundation is off, everything else will be too! After performing this division, we add 54 to the result. This step is straightforward but still requires careful attention to detail. Now, we've simplified the expression inside the parentheses.
Next up, we multiply the result from the parentheses by 6. Again, accuracy is paramount. Multiplication is one of those operations where it’s easy to make a small mistake that leads to a big problem. After this multiplication, we add 47928. This final addition gives us the total value of expression A. Now comes the moment of truth: is this final number divisible by 9? To find out, we add up all the digits in the result. If the sum of the digits is divisible by 9, then the entire number is divisible by 9. If not, then expression A does not meet our criterion. This divisibility rule is a neat trick that saves us from having to do long division!
By performing each step carefully and double-checking our work, we can confidently determine whether the value of expression A is divisible by 9. This process not only gives us the answer but also reinforces our understanding of arithmetic operations and divisibility rules. So, let’s crunch those numbers and see what we get!
Expression B Calculation and Divisibility Analysis
Now, let’s move on to expression B: . This one looks a bit different, but we'll apply the same careful approach. Our focus remains on determining whether the final result, after all operations are completed, is divisible by 9. Like before, we'll break it down step by step, ensuring we don’t miss any details. Precision is our watchword here.
We start with the division and multiplication inside the parentheses. First, we divide 720 by 24. This gives us a quotient, which we then multiply by 60. It's essential to perform these operations in the correct order to adhere to the order of operations (PEMDAS/BODMAS). Missteps here can lead to a completely wrong answer. After the multiplication, we add 33200 to the result. This addition gives us the value inside the parentheses.
Next, we divide the result from the parentheses by 7. This division is a critical step, and we need to ensure it’s done accurately. The quotient we obtain here will be used in the final subtraction. Finally, we subtract 3029 from the quotient. This gives us the final value of expression B. Now, we need to check if this final number is divisible by 9. Just like before, we sum the digits of the final result. If this sum is divisible by 9, then the number itself is divisible by 9. If it’s not, then expression B does not satisfy our condition.
By meticulously working through each step—division, multiplication, addition, and subtraction—we can arrive at the correct final value for expression B. Checking divisibility by 9 is the final piece of the puzzle, and it relies on the accuracy of all preceding steps. So, let’s get those calculations done and see if expression B passes the divisibility test!
Expression C Calculation and Divisibility Analysis
Alright, let’s tackle the final boss: expression C: . This expression involves division, subtraction, and multiplication, so we need to stay sharp. As with the previous expressions, our main goal is to determine if the final result is divisible by 9. We'll take it one step at a time, focusing on accuracy and clarity. Thoroughness is key to solving these problems.
First, we perform the division: 54009 divided by 9. This should give us a whole number since 54009 is likely chosen to be divisible by 9 for the sake of the problem. Next, we tackle the subtraction inside the parentheses: 960221 minus 957320. This gives us a difference that we will then multiply by 4. The subtraction and multiplication need to be done in the correct order according to the order of operations.
After the multiplication, we add the result to the quotient we obtained from the initial division. This final addition gives us the total value of expression C. Now, it’s time to check for divisibility by 9. We add up all the digits in the final result. If the sum is divisible by 9, then the entire number is divisible by 9, and expression C meets our criterion. If the sum is not divisible by 9, then expression C does not pass the test.
By carefully performing each operation and verifying our calculations, we can confidently determine whether expression C is divisible by 9. This step-by-step approach helps us avoid errors and ensures we understand the process thoroughly. So, let’s wrap up the calculations and see if expression C makes the cut!
Comprehensive Analysis and Divisibility Conclusion
After meticulously calculating expressions A, B, and C, we now need to consolidate our findings and draw a conclusion. For each expression, we performed a series of arithmetic operations, and then, most importantly, we checked whether the final result was divisible by 9. Remember, a number is divisible by 9 if the sum of its digits is divisible by 9. This is a handy rule that simplifies our task considerably.
To provide a comprehensive analysis, let’s recap the key steps for each expression. For expression A, we handled multiplication, division, addition, and another multiplication, followed by a final addition. For expression B, we performed division, multiplication, addition, another division, and finally, subtraction. Expression C involved division, subtraction, multiplication, and addition. Each of these steps had to be executed with precision to ensure the final result was accurate.
Now, based on our calculations, we have a final value for each expression. To determine if all values are divisible by 9, we apply the divisibility rule to each final result. If the sums of the digits for all three expressions are divisible by 9, then our initial question is answered affirmatively. If even one expression fails this test, then the answer is no.
This exercise underscores the importance of methodical calculation and the practical application of divisibility rules. It also highlights how complex arithmetic problems can be broken down into manageable steps. So, what’s the final verdict? Are all the values divisible by 9? Let’s crunch those final numbers and find out!
In conclusion, by methodically working through each expression and applying the divisibility rule for 9, we can confidently determine whether all calculated values are divisible by 9. This exercise not only tests our arithmetic skills but also reinforces our understanding of fundamental mathematical principles. Great job, guys, for sticking with it till the end!
