Divisibility Rules Explained Is 87 Divisible By 6

Hey there, math enthusiasts! Let's dive into some divisibility rules and explore whether the numbers mentioned are divisible by the given divisors. We'll break down each case, making it super easy to understand. We will analyze whether 87 is divisible by 6, investigate the divisibility of 43 by 4 and 8, check if 30_ is divisible by 1000, examine the divisibility of 63 by 5 and 3, and finally, determine if 93 is divisible by 5 but not by 10. So, grab your thinking caps, and let's get started!

Is 87 Divisible by 6?

When we're trying to figure out if a number is divisible by 6, we're really asking if it can be divided evenly by both 2 and 3. Why? Because 6 is the product of 2 and 3. This is a fundamental concept in number theory, and it makes our lives much easier when dealing with divisibility questions. So, let's break down 87 and see if it fits the bill. The divisibility rules are our best friends here. A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). A number is divisible by 3 if the sum of its digits is divisible by 3. These simple rules save us from doing long division every single time, and they're incredibly handy in various math problems. For 87, we first check if it's divisible by 2. The last digit is 7, which is not an even number. Right away, we know that 87 is not divisible by 2. But just for kicks, let's also check the divisibility rule for 3. The sum of the digits in 87 is 8 + 7 = 15. Now, is 15 divisible by 3? Yes, it is! 15 ÷ 3 = 5. So, 87 is divisible by 3, but since it's not divisible by 2, it can't be divisible by 6. To be divisible by 6, a number has to pass both tests. Think of it like needing two keys to open a lock; if you only have one key, you're stuck. In conclusion, 87 is not divisible by 6 because it doesn't meet the divisibility criteria for both 2 and 3. This simple check highlights how understanding divisibility rules can quickly help us determine factors and multiples of numbers. Divisibility rules aren't just cool tricks; they are essential tools in prime factorization, simplifying fractions, and solving a plethora of mathematical problems. This is the backbone of understanding how numbers interact, and it's a concept that will stick with you through more advanced math topics. So, let's move on to the next number and keep those divisibility skills sharp!

Is 43 Divisible by 4 but Not by 8?

Now, let's investigate the divisibility of 43 by 4 and 8. These numbers are closely related, but the rules for divisibility are slightly different. Understanding these differences is crucial for mastering divisibility. So, is 43 divisible by 4? To check this, we look at the last two digits of the number. If those digits form a number that is divisible by 4, then the whole number is divisible by 4. In the case of 43, the last two digits are simply 43. We need to determine if 43 is divisible by 4. You guys probably already know that 43 divided by 4 is 10 with a remainder of 3. So, 43 is not divisible by 4. Therefore, the statement that 43 is divisible by 4 is incorrect. This initial check is vital because it sets the stage for the rest of our analysis. If a number isn’t divisible by a smaller factor, it certainly won’t be divisible by a larger multiple of that factor. Now, let's consider the second part of the question: is 43 not divisible by 8? To check divisibility by 8, we need to look at the last three digits of the number. If these digits form a number that is divisible by 8, then the whole number is divisible by 8. Since 43 is a two-digit number, we can simply consider the number 43 itself. Let's divide 43 by 8. 43 divided by 8 is 5 with a remainder of 3. So, 43 is not divisible by 8. This confirms the second part of the statement. Here, we see a clear difference in how divisibility rules work for different numbers. While the divisibility rule for 4 involves checking the last two digits, the rule for 8 involves checking the last three digits. This might seem like a small detail, but it's essential for accurate calculations. To sum it up, 43 is not divisible by 4, and it is also not divisible by 8. So, the original statement that 43 is divisible by 4 but not by 8 is partly incorrect. This exercise underscores the importance of carefully applying divisibility rules and double-checking our work to avoid errors. It also reinforces the idea that each divisibility rule is specific to the number in question, and using the right rule is key to getting the right answer. Understanding these nuances makes us better at handling more complex mathematical problems down the line.

Is 30_ Divisible by 1000?

Next up, we're tackling the question of whether 30_ (where the underscore represents a missing digit) is divisible by 1000. This is an interesting one because it involves a little bit of detective work. Divisibility by 1000 has a straightforward rule: a number is divisible by 1000 if its last three digits are zeros. Think about it – 1000, 2000, 3000, and so on are all divisible by 1000. It's a pattern that's easy to spot once you know what to look for. So, for 30_ to be divisible by 1000, the last three digits need to be 000. We already have 30 as the first two digits. This means the missing digit, represented by the underscore, must be a 0 for the number to be divisible by 1000. Let's fill in the blank: if the number is 3000, then it is divisible by 1000. 3000 ÷ 1000 = 3, with no remainder. Perfect! But what if the missing digit were anything else? If it were 1, making the number 301, that number is definitely not divisible by 1000. The same goes for any other digit from 1 to 9. The number must end in three zeros to meet the divisibility rule for 1000. This exercise illustrates how divisibility rules can help us fill in missing information or confirm the properties of numbers. It's not just about dividing numbers; it's about understanding the structure of numbers themselves. When we look at divisibility, we're really exploring how numbers are built and how they relate to each other. In this case, the rule for 1000 is a great example of how powers of 10 (10, 100, 1000, etc.) have simple divisibility criteria based on the number of zeros. This understanding is invaluable when you're working with large numbers, scientific notation, or any situation where place value is crucial. So, to recap, 30_ is divisible by 1000 only if the missing digit is 0, making the number 3000. This simple yet powerful rule helps us quickly determine the divisibility and highlights the importance of place value in our number system.

Is 63 Divisible by 5 and by 3?

Let’s explore whether 63 is divisible by both 5 and 3. This requires us to apply two different divisibility rules and see if 63 meets both criteria. Think of it as a double-check system for numbers. To start, let's consider the divisibility rule for 5. A number is divisible by 5 if its last digit is either 0 or 5. This is a straightforward rule that's easy to remember. Now, let's look at 63. The last digit is 3, which is neither 0 nor 5. Therefore, 63 is not divisible by 5. This immediately tells us that the entire statement—63 is divisible by both 5 and 3—is incorrect. Even if it were divisible by 3, it wouldn't matter because it fails the divisibility test for 5. But, for the sake of completeness, let's also check the divisibility by 3. The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. So, we add the digits of 63: 6 + 3 = 9. Is 9 divisible by 3? Yes, it is! 9 ÷ 3 = 3. So, 63 is divisible by 3. We've now confirmed that 63 is divisible by 3 but not by 5. This exercise is a perfect illustration of why it's crucial to check all conditions in a mathematical statement. Just because a number satisfies one condition doesn't mean it satisfies them all. In this case, 63 passed the divisibility test for 3 but failed the test for 5. Understanding this principle is vital in more complex math scenarios where multiple conditions must be met for a solution to be valid. It also reinforces the idea that each divisibility rule provides specific insights into the properties of numbers. When we apply these rules, we're not just doing arithmetic; we're gaining a deeper understanding of how numbers are structured and how they interact. In summary, 63 is not divisible by 5, but it is divisible by 3. The original statement, claiming that 63 is divisible by both 5 and 3, is therefore false. This kind of analysis helps us to be precise and accurate in our mathematical thinking.

Is 93 Divisible by 5 but Not by 10?

Finally, let's tackle the question: is 93 divisible by 5 but not by 10? This involves two divisibility rules, similar to our previous example, but with a slight twist. We need to check not only if 93 is divisible by 5 but also if it's not divisible by 10. First, let's check the divisibility rule for 5. As we know, a number is divisible by 5 if its last digit is either 0 or 5. Looking at 93, the last digit is 3, which is neither 0 nor 5. Therefore, 93 is not divisible by 5. This immediately gives us a clue about the statement's accuracy. Since 93 is not divisible by 5, the first part of the statement is false. However, let's continue to the second part to see if 93 is indeed not divisible by 10, just to be thorough. The divisibility rule for 10 is quite simple: a number is divisible by 10 if its last digit is 0. The last digit of 93 is 3, not 0. Thus, 93 is not divisible by 10. So, while the second part of the statement is correct—93 is not divisible by 10—the first part is incorrect. The overall statement claims that 93 is divisible by 5 but not by 10. Since 93 is not divisible by 5, the entire statement is false. This highlights the importance of carefully analyzing each part of a statement and ensuring that all conditions are met for the statement to be true. In mathematical logic, the word "but" often implies a conjunction, meaning both parts of the statement must be true. This exercise also reinforces the idea that divisibility rules are powerful tools for quickly assessing the properties of numbers. They allow us to make judgments about factors and multiples without performing long division, which saves time and reduces the chance of errors. Understanding the divisibility rules for 5 and 10, in particular, is helpful in everyday situations, such as dividing quantities or checking the accuracy of calculations. In conclusion, the statement that 93 is divisible by 5 but not by 10 is false because 93 is not divisible by 5. This emphasizes the need for precise thinking and careful application of mathematical rules.

We've had quite the mathematical workout today, exploring various divisibility rules and number properties. Remember, mastering these concepts is super helpful for tackling all sorts of math problems. Keep practicing, and you'll become a divisibility pro in no time!