Divisibility By 3 And 5: Find The Numbers!

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Hey guys! Let's dive into the world of numbers and divisibility. We've got a list of numbers here: 48, 58, 180, 27900, 63672, 42324, and 34410. Our mission, should we choose to accept it, is to find out which of these numbers are divisible by both 3 and 5. Sounds like a fun math puzzle, right? Let's break it down step by step so it's super clear.

Understanding Divisibility Rules

Before we jump into the numbers, let's quickly refresh our memory on the divisibility rules for 3 and 5. These rules are like little shortcuts that make our lives much easier. Think of them as our secret weapons in this mathematical quest!

Divisibility Rule for 3

The divisibility rule for 3 is pretty neat. A number is divisible by 3 if the sum of its digits is divisible by 3. So, what does that mean in simple terms? Well, let's say we have a number like 123. To check if it's divisible by 3, we add up its digits: 1 + 2 + 3 = 6. Since 6 is divisible by 3, then 123 is also divisible by 3. See? Easy peasy!

We need to remember this rule because it's super important for finding the numbers that fit our criteria. It's like having a special decoder ring for the number 3. Keep this rule in the back of your mind as we move forward. We'll be using it a lot!

Divisibility Rule for 5

Now, let's talk about the divisibility rule for 5. This one is even simpler. A number is divisible by 5 if its last digit is either 0 or 5. That’s it! No complicated calculations, no adding up digits. Just a quick glance at the last number, and we're good to go.

For example, if we have the number 45, the last digit is 5, so it's divisible by 5. If we have 100, the last digit is 0, so it’s also divisible by 5. If we have 72, the last digit is 2, so it’s not divisible by 5. See how straightforward that is? This rule is our second secret weapon in identifying the right numbers. Remember, we're looking for numbers that are divisible by both 3 and 5, so we need to keep both rules in mind.

Applying the Rules to Our Numbers

Alright, now for the fun part! Let's take our divisibility rule knowledge and apply it to the list of numbers we have: 48, 58, 180, 27900, 63672, 42324, and 34410. We’re going to go through each number, one by one, and see if it passes both the divisibility tests for 3 and 5. It’s like being a number detective, and these rules are our magnifying glass.

Checking Each Number

Let’s roll up our sleeves and get started! We’ll take each number and put it through the wringer, checking if it’s divisible by 3 and 5.

  1. 48: First, let’s check for divisibility by 3. The sum of the digits is 4 + 8 = 12. Since 12 is divisible by 3, 48 is divisible by 3. Now, let's check for divisibility by 5. The last digit is 8, which is neither 0 nor 5, so 48 is not divisible by 5. So, 48 doesn't make the cut.
  2. 58: Time for 58! For divisibility by 3, the sum of the digits is 5 + 8 = 13. 13 is not divisible by 3, so 58 isn't divisible by 3 either. We don't even need to check for divisibility by 5 since it already failed the first test. Poor 58!
  3. 180: Now we're talking! Let's check 180. For divisibility by 3, the sum of the digits is 1 + 8 + 0 = 9. 9 is divisible by 3, so 180 passes the first test. For divisibility by 5, the last digit is 0, which means it’s divisible by 5. Woohoo! 180 is a winner!
  4. 27900: This looks like a big one, but let’s not be intimidated. For divisibility by 3, the sum of the digits is 2 + 7 + 9 + 0 + 0 = 18. 18 is divisible by 3, so 27900 is divisible by 3. For divisibility by 5, the last digit is 0, so it’s also divisible by 5. Double score! 27900 is another one that fits the bill.
  5. 63672: Let’s keep the momentum going. For divisibility by 3, the sum of the digits is 6 + 3 + 6 + 7 + 2 = 24. 24 is divisible by 3, so 63672 is divisible by 3. For divisibility by 5, the last digit is 2, which means it’s not divisible by 5. Better luck next time, 63672.
  6. 42324: Almost there! For divisibility by 3, the sum of the digits is 4 + 2 + 3 + 2 + 4 = 15. 15 is divisible by 3, so 42324 is divisible by 3. But, for divisibility by 5, the last digit is 4, so it’s not divisible by 5. Close, but no cigar for 42324.
  7. 34410: Last but not least, let’s check 34410. For divisibility by 3, the sum of the digits is 3 + 4 + 4 + 1 + 0 = 12. 12 is divisible by 3, so 34410 is divisible by 3. And for divisibility by 5, the last digit is 0, which means it’s divisible by 5. We have another winner! 34410 joins the club.

Conclusion: The Numbers Divisible by Both 3 and 5

So, after our number detective work, we’ve found the numbers that are divisible by both 3 and 5. Drumroll, please! The numbers are 180, 27900, and 34410. Great job, team!

Isn't it cool how these simple divisibility rules can help us solve problems quickly? Math can be like a fun puzzle if we know the tricks. Keep these rules in your back pocket, and you’ll be a divisibility pro in no time! Whether you're tackling homework, prepping for a test, or just curious about numbers, these skills will definitely come in handy. Keep exploring the world of math, guys! There’s always something new and exciting to discover.