Finding The Remainder Of 2n Divided By 12 When N Divided By 12 Has A Remainder Of 7

Hey guys! Let's dive into a cool math problem today. We're going to figure out how to find remainders when we're dividing numbers. This might sound a bit tricky, but trust me, it's super interesting once you get the hang of it. We'll break it down step by step, so you can follow along easily. So, grab your thinking caps, and let's get started!

Understanding the Problem: Remainders and Division

Before we jump into solving the specific problem, let's make sure we're all on the same page about what remainders are and how they work in division. When you divide one number by another, sometimes the division is perfect, and you get a whole number as the result. But other times, you have something left over – that leftover is what we call the remainder. Think of it like sharing cookies among friends. If you have 10 cookies and 5 friends, each friend gets 2 cookies, and there are no cookies left over. But if you have 11 cookies and 5 friends, each friend still gets 2 cookies, but you have 1 cookie left over. That 1 cookie is the remainder.

In mathematical terms, when we divide a number n by another number d, we can write it like this:

n = q * d + r

Where:

• n is the dividend (the number being divided)
• d is the divisor (the number we're dividing by)
• q is the quotient (the whole number result of the division)
• r is the remainder (the amount left over)

For example, if we divide 17 by 5:

17 = 3 * 5 + 2

Here, the quotient (q) is 3, and the remainder (r) is 2. So, understanding this basic concept of division and remainders is super crucial for tackling the problem at hand. We'll be using this idea throughout our solution, so make sure you're comfortable with it. It's like having the right tools before you start a project – it makes everything so much easier!

Setting Up the Problem: n Divided by 12

Now, let's get specific and look at the problem we're trying to solve. We're told that when a number n is divided by 12, the remainder is 7. How do we write that mathematically? Well, using the same formula we talked about earlier, we can express this as:

n = 12 * q + 7

Here, q represents the quotient, which is the whole number result we get when we divide n by 12. The key thing to notice here is that the remainder is 7. This means that n is 7 more than some multiple of 12. Think of it like this: n is a number that, if you took away 7, would be perfectly divisible by 12. This is a really important piece of information, and writing it down like this helps us see the structure of the number n. It's like having a secret code that unlocks the solution!

To really nail this down, let's think about a few examples. If q is 1, then n would be 12 * 1 + 7 = 19. If q is 2, then n would be 12 * 2 + 7 = 31. If q is 3, then n would be 12 * 3 + 7 = 43. Notice how all these numbers (19, 31, 43) leave a remainder of 7 when divided by 12. This gives us a concrete way to think about what n could be, and it sets the stage for the next part of the problem, where we'll be looking at 2n. So, remember, writing the given information in a mathematical form is the first big step in solving this problem. It's like laying the foundation for a building – you need a solid base to build on!

Finding 2n: Doubling the Number

Okay, so we know that n can be written as 12 * q + 7. Now, the next part of our problem asks us to find the remainder when 2n is divided by 12. So, what's the first thing we need to do? That's right, we need to figure out what 2n actually is. To do this, we simply multiply both sides of our equation by 2:

2 * n = 2 * (12 * q + 7)

Now, we use the distributive property (remember that from algebra?) to multiply the 2 across the parentheses:

2 * n = 2 * 12 * q + 2 * 7

This simplifies to:

2 * n = 24 * q + 14

So, now we have an expression for 2n. It's 24 times q, plus 14. This is a super important step because it gives us a way to work with 2n in terms of numbers we already understand. It's like translating a sentence into a language you know – once you understand the words, you can start to make sense of the whole thing!

Think about it this way: 2n is made up of two parts. The first part, 24 * q, is a multiple of 24 (and therefore also a multiple of 12, since 24 is 2 times 12). The second part is 14. This is going to be key to finding the remainder when we divide 2n by 12. We've essentially doubled our original number n, and now we need to see how that affects the remainder. So, we're one step closer to cracking this problem! Stick with it, guys, we're doing great!

Determining the Remainder: Dividing 2n by 12

Alright, we've got 2n expressed as 24q + 14. Now, our mission is to find the remainder when we divide 2n by 12. Remember, we're looking for that leftover bit after we've divided as much as we can by 12. This is where our understanding of remainders really comes into play.

Let's break down 2n piece by piece. We have 24q + 14. The first part, 24q, is super interesting because 24 is a multiple of 12. In fact, 24 is exactly 2 times 12. So, we can rewrite 24q as 2 * 12 * q. This means that 24q is perfectly divisible by 12, no matter what q is. Think of it like having a big box of cookies that's a multiple of 12 – you can share them equally among 12 friends with no leftovers.

So, when we divide 24q by 12, we get a whole number (2q) and no remainder. That's awesome because it means the remainder when we divide 2n by 12 is going to come entirely from the second part, the 14. Now we just need to figure out what happens when we divide 14 by 12. This is much simpler, right?

When we divide 14 by 12, we get a quotient of 1 and a remainder of 2. This is because 14 can be written as 1 * 12 + 2. So, the remainder is 2. And guess what? That's the remainder when we divide 2n by 12! We've cracked it!

The Answer: Remainder of 2

So, guys, after all that awesome math work, we've found our answer! The remainder when 2n is divided by 12 is 2. How cool is that? We took a problem that might have seemed a bit mysterious at first, and we broke it down into smaller, manageable steps. We used our understanding of remainders, division, and a little bit of algebra to get to the solution. This is what math is all about – using the tools and concepts you know to solve new and interesting problems.

Let's quickly recap what we did. We started with the information that n divided by 12 has a remainder of 7. We wrote this mathematically as n = 12q + 7. Then, we found 2n by multiplying both sides of the equation by 2, giving us 2n = 24q + 14. Finally, we realized that 24q is divisible by 12, so the remainder when 2n is divided by 12 is the same as the remainder when 14 is divided by 12, which is 2. And there you have it! A clear, step-by-step solution to our problem.

I hope this explanation helped you understand how to tackle problems involving remainders. Remember, the key is to break things down, write them mathematically, and use the tools you have to find the solution. Keep practicing, and you'll become a remainder-solving pro in no time! Keep up the great work, and I'll catch you in the next math adventure!