Divisibility Rules: Finding The Product Of Divisors Of 20,070

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Hey guys! Ever stumbled upon a math problem that seems like a puzzle? Today, we're diving deep into the world of divisibility to crack one such puzzle. We've got a question that mixes divisibility rules with a bit of multiplication, and it's super intriguing. So, let's put on our thinking caps and get started!

The Divisibility Challenge: What's the Product?

Our main divisibility challenge is this: Based on divisibility criteria, what is the product of the natural numbers less than 10 that divide the number 20,070? We have five options to choose from: A. 1,620, B. 180, C. 11,340, D. 6,480, and E. 16,200. It might seem daunting at first, but don't worry! We'll break it down step by step, making sure we understand every part of the process. To tackle this, we need to understand divisibility rules and how they help us identify factors of a number quickly. We need to identify all the natural numbers less than 10 that divide 20,070 without leaving a remainder. Then, once we've identified these numbers, we simply multiply them together to find the product. It’s like a mini treasure hunt where the treasure is the correct product! Remember, natural numbers are the positive whole numbers (1, 2, 3, and so on). This means we're looking for numbers from 1 to 9 that divide 20,070 perfectly. The beauty of this problem lies in its practical approach. Divisibility isn't just a theoretical concept; it's something we use in everyday life, from dividing a pizza equally among friends to organizing items into groups. By mastering these concepts, we're not just solving a math problem; we're building valuable problem-solving skills that will help us in all sorts of situations. So, let’s roll up our sleeves and get ready to explore the fascinating world of divisibility!

Cracking the Code: Divisibility Rules to the Rescue

To efficiently solve our divisibility challenge, we need to harness the power of divisibility rules. These rules are like secret shortcuts that tell us whether a number is divisible by another number without actually doing the division. They save us time and effort, making the whole process much smoother. Let's quickly recap the divisibility rules for numbers 2 through 9, as these are the natural numbers less than 10 that we need to consider.

  • Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5.
  • Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • We don't have a divisibility rule for 7 because the most straightforward method to determine divisibility by 7 is direct division.
  • There isn’t a separate divisibility rule for 8 in this particular list because it's not immediately necessary for solving the problem, but generally, a number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Now that we have our toolkit ready, let's apply these rules to the number 20,070 and see which numbers less than 10 are its divisors. This is where the fun begins – it's like being a detective, using clues to solve a mystery! So, let's put these rules into action and unveil the divisors of 20,070.

The Hunt for Divisors: Applying the Rules to 20,070

Now, let's get our hands dirty and apply these divisibility rules to our number, 20,070. This is where the magic happens! We'll go through each number from 1 to 9 and see if it divides 20,070. Remember, 1 divides every number, so that's an easy one to start with.

  • Divisibility by 1: Of course, 1 divides 20,070.
  • Divisibility by 2: The last digit of 20,070 is 0, which is even, so it is divisible by 2.
  • Divisibility by 3: The sum of the digits of 20,070 is 2 + 0 + 0 + 7 + 0 = 9, which is divisible by 3, so 20,070 is divisible by 3.
  • Divisibility by 4: The last two digits of 20,070 are 70, which is not divisible by 4, so 20,070 is not divisible by 4.
  • Divisibility by 5: The last digit of 20,070 is 0, so it is divisible by 5.
  • Divisibility by 6: Since 20,070 is divisible by both 2 and 3, it is also divisible by 6.
  • Divisibility by 7: To check for divisibility by 7, we'll perform the division: 20,070 ÷ 7 = 2,867 with a remainder of 1. So, 20,070 is not divisible by 7.
  • Divisibility by 8: The divisibility rule for 8 involves checking if the last three digits of the number are divisible by 8. In this case, the last three digits are 070, which is not divisible by 8 (70 / 8 = 8 with a remainder of 6). Thus, 20,070 is not divisible by 8.
  • Divisibility by 9: The sum of the digits is 9, which is divisible by 9, so 20,070 is divisible by 9.

So, the numbers less than 10 that divide 20,070 are 1, 2, 3, 5, 6, and 9. We've identified our divisors – now it's time for the final step: finding the product of these numbers. It’s like the grand finale of our mathematical investigation!

The Grand Finale: Multiplying the Divisors

Alright, we've reached the final stage of our divisibility adventure! We've successfully identified the natural numbers less than 10 that divide 20,070. These numbers are 1, 2, 3, 5, 6, and 9. Now, the last step is to find the product of these divisors. This is where our multiplication skills come into play. Let's multiply these numbers together: 1 * 2 * 3 * 5 * 6 * 9.

We can break this down step by step to make it easier. First, let's multiply the smaller numbers: 1 * 2 = 2. Then, 2 * 3 = 6. Now we have 6 * 5 * 6 * 9. Let's continue: 6 * 5 = 30. Now we have 30 * 6 * 9. Next, 30 * 6 = 180. Finally, we multiply 180 * 9. To calculate 180 * 9, we can think of it as (100 * 9) + (80 * 9), which is 900 + 720 = 1,620.

So, the product of the natural numbers less than 10 that divide 20,070 is 1,620. We've done it! We've cracked the code and found our answer. This final step ties everything together and gives us the solution we were looking for. It’s like completing a puzzle – the satisfaction of seeing all the pieces fit perfectly is truly rewarding!

Solution and Reflection: Choosing the Correct Answer

After our meticulous calculation, we've arrived at the answer: the product of the natural numbers less than 10 that divide 20,070 is 1,620. Now, let's look back at our options and choose the correct one.

  • A. 1,620
  • B. 180
  • C. 11,340
  • D. 6,480
  • E. 16,200

Clearly, the correct answer is A. 1,620. We've successfully navigated through the divisibility rules, identified the divisors, and multiplied them together to find the solution. This journey through the world of divisibility has been both challenging and rewarding. We've not only solved a specific problem but also reinforced our understanding of fundamental mathematical concepts. Divisibility rules are powerful tools that help us simplify complex calculations and solve problems more efficiently. By mastering these rules, we're equipped to tackle a wide range of mathematical challenges with confidence. So, congratulations on making it to the end! You've not only found the answer but also deepened your understanding of divisibility. Keep practicing, keep exploring, and most importantly, keep enjoying the beauty of mathematics!

Based on the criteria of divisibility, what is the product of the natural numbers less than 10 that divide the number 20,070? Options: A. 1,620, B. 180, C. 11,340, D. 6,480, E. 16,200.

Divisibility Rules Find Product of Divisors of 20070