Math Help: Find Number Divisible By 7 & Divisor Of 63
Hey there, future mathematicians! Let's tackle this interesting math problem together. It seems like we need to find a number that fits a few specific rules. This type of problem is all about understanding multiples, divisors, and working within a range. Don't worry; we'll break it down step by step so it's super easy to understand. We are going to look into the core concepts first. Think of it like this: we're detectives trying to find a mystery number, and these rules are our clues. We've got to be both a multiple of 7, a divisor of 63, and fit nicely between 10 and 50. It sounds like a puzzle, right? Math can be like that, a fun puzzle to solve! Understanding multiples and divisors is key here, because a multiple is what you get when you multiply a number by an integer (like 1, 2, 3, and so on), while a divisor is a number that divides evenly into another number. We also have the range constraint, meaning the number has to be between 10 and 50. This helps narrow down the possibilities, so we don't have to search through every number in the universe.
Understanding Multiples and Divisors
First, let's talk about what it means to be a multiple of 7. Guys, a multiple of 7 is simply any number you get when you multiply 7 by a whole number. Think of it like the 7 times table! So, 7 x 1 = 7, 7 x 2 = 14, 7 x 3 = 21, and so on. We need to keep this in mind because our mystery number has to be somewhere in this sequence. Now, what about being a divisor of 63? A divisor is a number that divides evenly into another number, meaning there's no remainder. For example, 7 is a divisor of 63 because 63 divided by 7 is exactly 9. But 8 isn't a divisor of 63 because when you divide 63 by 8, you get a remainder. We need a number that not only shows up in the 7 times table but also divides 63 perfectly. To find the divisors of 63, we can think about what numbers multiply together to give us 63. We know 1 x 63 = 63, 3 x 21 = 63, and 7 x 9 = 63. So, the divisors of 63 are 1, 3, 7, 9, 21, and 63. See how understanding these definitions helps us narrow down our options? We're building a solid foundation for solving the problem, and that's super important in math. It's not just about getting the answer, it's about understanding why the answer is what it is. This foundation will help you tackle tougher problems later on!
Finding Multiples of 7
Okay, let's get specific and list out the multiples of 7 to help us find our mystery number. Remember, we're looking for a number between 10 and 50, so we don't need to go too crazy. Let's start with the basics: 7 x 1 = 7, which is too small. 7 x 2 = 14. That's a contender! 7 x 3 = 21. Another possibility! We're on a roll, guys! Let's keep going: 7 x 4 = 28, 7 x 5 = 35, 7 x 6 = 42, and 7 x 7 = 49. All of these are within our range. Now, 7 x 8 = 56, which is too big, so we can stop there. So far, we have the following multiples of 7 between 10 and 50: 14, 21, 28, 35, 42, and 49. That's a good list to work with! But remember, our mystery number isn't just any multiple of 7; it also needs to be a divisor of 63. This is where our detective work gets even more interesting! We've narrowed down the possibilities a lot already, and we're using our knowledge of multiples to do it. It's like we're filtering out the wrong numbers, getting closer and closer to the right one. Listing out the multiples like this makes the problem much more manageable. We can see the numbers clearly, and it becomes easier to check if they fit our other rules.
Identifying Divisors of 63
Alright, let's switch gears and focus on the divisors of 63. Remember, a divisor of 63 is a number that divides evenly into 63. We already touched on this earlier, but let's make a complete list to make sure we've got all our options. We need to ask ourselves: what numbers can we divide 63 by without getting a remainder? 1 is always a divisor of any number, so 1 is a divisor of 63. 2 doesn't divide evenly into 63 (try it!), so it's not a divisor. 3 does divide evenly into 63 (63 / 3 = 21), so 3 is a divisor. 4, 5, and 6 don't divide evenly into 63. But 7 does! (63 / 7 = 9). So 7 is another divisor. 8 doesn't divide evenly, but 9 does (63 / 9 = 7). So 9 is on our list too. Now, here's a little trick: once you get past the square root of the number (which is a little more than 7 for 63), you're just finding the pairs of the divisors you already found. For example, since 7 is a divisor and 63 / 7 = 9, then 9 is also a divisor. Continuing, we find that 21 (63 / 3 = 21) and 63 (63 / 1 = 63) are also divisors. So, the complete list of divisors of 63 is: 1, 3, 7, 9, 21, and 63. Excellent! Now we have two lists: multiples of 7 between 10 and 50, and divisors of 63. The next step is to compare these lists and see if there are any numbers that appear on both.
Finding the Common Number
Okay, guys, this is where the magic happens! We've got our list of multiples of 7 between 10 and 50: (14, 21, 28, 35, 42, 49). And we have our list of divisors of 63: (1, 3, 7, 9, 21, 63). Now, let's put on our detective hats and see if we can spot any numbers that are on both lists. This is like a Venn diagram in action! Are there any numbers that belong to both the "multiples of 7" club and the "divisors of 63" club? Scan through the lists carefully. Do you see any matches? Aha! I see one! The number 21 appears on both lists. That's a very promising sign! But wait, there's another one! The number 7 is a divisor of 63, but it is not between 10 and 50, so it doesn't match the condition. Let's focus on 21 for a moment. It's a multiple of 7 (7 x 3 = 21), and it's a divisor of 63 (63 / 21 = 3). It seems like 21 is fitting all the clues so far! But let's not jump to conclusions just yet. We need to make absolutely sure it meets all the requirements. We know it's a multiple of 7 and a divisor of 63. But is it within the range of 10 and 50? Yes, it is! 21 is definitely bigger than 10 and smaller than 50. So, drumroll please... it looks like we've found our mystery number! But let's just double-check one more time, just to be super sure.
The Solution
So, let's recap. We were on a mission to find a number that is a multiple of 7, a divisor of 63, and between 10 and 50. We methodically listed out the multiples of 7 within our range, and we identified all the divisors of 63. Then, we compared our lists and found that the number 21 appears on both. It's a multiple of 7 (7 x 3 = 21), it's a divisor of 63 (63 / 21 = 3), and it falls neatly between 10 and 50. Therefore, the answer to our math puzzle is 21! Awesome job, everyone! We cracked the case! Remember, math problems like this are all about breaking things down into smaller, more manageable steps. We used our knowledge of multiples, divisors, and ranges to solve this problem. And the best part is, we didn't just find the answer; we understood why the answer is what it is. That's the real power of math! You've got the skills to tackle similar problems now, so keep practicing and keep exploring the wonderful world of numbers!
I hope this explanation helps you understand the problem and the solution clearly. If you have any more math questions, don't hesitate to ask! Keep up the great work! Math is fun, and you're doing an amazing job learning! Never be afraid to ask for help, because that's how we all learn and grow. You've got this!