Numbers Divisible By 5 & 9: A Math Guide

Hey math enthusiasts! Let's dive into a cool number theory problem. We're going to figure out how to find numbers that are divisible by both 5 and 9. Specifically, we're looking at numbers in the format a7b. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making it super easy to understand. So, grab your pencils and let's get started!

Understanding the Basics: Divisibility Rules

Before we jump into the problem, let's quickly recap the divisibility rules for 5 and 9. These rules are your best friends in this kind of problem.

• Divisibility by 5: A number is divisible by 5 if its last digit is either 0 or 5. Easy peasy, right? This means that in our number a7b, the digit b must be either 0 or 5.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 81 is divisible by 9 because 8 + 1 = 9, and 9 is divisible by 9. Similarly, the number 126 is divisible by 9 because 1 + 2 + 6 = 9. Keep this in mind as we are going to use it.

So, with these rules in mind, we can start to decode our mystery number a7b.

Now we're ready to put these rules into action. This process will help you grasp the underlying mathematical concepts and approach similar problems with confidence. It's all about logical deduction and a little bit of number sense. This is a game of deduction, and you, my friend, are the detective. Each rule is a clue, and we're piecing together the puzzle to find the solution. Let's start with the first step which will unveil the secrets of the number. The cool thing about math is that once you understand the core principles, you can apply them to solve all sorts of problems. It's all connected. Understanding the basics will make the rest of the problem-solving much smoother. You are building a strong foundation for future problems.

Step-by-Step Solution: Unveiling a7b

Alright, let's get down to business and find the numbers a7b that fit our criteria. We'll consider the two possibilities for b based on the divisibility rule of 5.

Case 1: b = 0

If b = 0, our number becomes a70. Now, for this number to be divisible by 9, the sum of its digits (a + 7 + 0) must be divisible by 9. So, a + 7 must be divisible by 9.

The possible values for a are single digits (0 to 9). Let's see which values of a make a + 7 a multiple of 9.

• If a = 2, then 2 + 7 = 9 (which is divisible by 9). So, one solution is 270.

Therefore, when b = 0, the only number that satisfies the conditions is 270. Great job, guys!

Case 2: b = 5

If b = 5, our number becomes a75. For this number to be divisible by 9, the sum of its digits (a + 7 + 5) must be divisible by 9. That means a + 12 must be divisible by 9.

Let's find the values of a that make a + 12 a multiple of 9.

• If a = 6, then 6 + 12 = 18 (which is divisible by 9). So, another solution is 675.

Therefore, when b = 5, the number that satisfies the conditions is 675. You're doing awesome!

In both cases, we systematically applied the divisibility rules, and we've now found all the numbers that fit the bill. The process we used is the same process you can use for other number puzzles, with minor adjustments. The core principles remain consistent. You’re building up a skill set that goes beyond just math; you're developing critical thinking abilities. With each problem, you're becoming a sharper, more analytical thinker. It's a fantastic exercise for your brain, and it's super rewarding when you solve a problem.

The Final Answer: The Numbers We've Found

So, after all that detective work, we've found our answers! The numbers a7b that are divisible by both 5 and 9 are:

• 270
• 675

And there you have it! We've successfully found all the numbers that meet the criteria. Isn't it satisfying when you solve a problem, especially when it involves numbers? Remember, the key is to break down the problem into smaller, manageable steps. Apply the rules systematically, and you'll get there. Every step we take brings us closer to a deeper understanding of number theory. And understanding is the real victory, isn't it? The ability to think critically and apply mathematical principles to solve problems is an incredibly valuable skill that extends far beyond the classroom.

Conclusion: You've Got This!

Congratulations, guys! You've successfully navigated the world of divisibility and solved the problem. You've seen how easy it is when you know the rules and follow a step-by-step approach. Keep practicing, keep exploring, and you'll become a master of number theory in no time. You have now acquired skills that you can use in a variety of situations. Remember, math is like a muscle; the more you exercise it, the stronger it gets. Each problem you solve makes you more confident and capable. Believe in your abilities, and you'll achieve great things! Keep up the amazing work!

This whole process may seem complicated at first, but with practice, it becomes second nature. Each problem you tackle builds your confidence and skills. Remember, the journey is as important as the destination. Embrace the challenges, learn from your mistakes, and celebrate your successes. You've got this, and I'm proud of the work you've put into understanding this. Keep up the excellent work, and enjoy the journey of learning. You're doing great, and I'm here to support you every step of the way!