Smallest 3-Digit Number With Remainder 5: Explained

by Admin 52 views
Finding the Smallest 3-Digit Number with Remainder 5 When Divided by 12 and 18

Hey guys! Let's tackle this math problem together. We're looking for the smallest three-digit number that leaves a remainder of 5 when divided by both 12 and 18. Sounds tricky? Don't worry, we'll break it down step by step.

Understanding the Problem

First, let's make sure we understand what the problem is asking. We need a number that:

  • Is a three-digit number (meaning it's between 100 and 999).
  • When divided by 12, it leaves a remainder of 5.
  • When divided by 18, it also leaves a remainder of 5.

So, essentially, if we subtract 5 from our target number, the result should be divisible by both 12 and 18. This is a crucial piece of information that will guide us to the solution.

Key Concepts: LCM and Remainders

To solve this, we'll need two important math concepts:

  1. Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both. In our case, we need to find the LCM of 12 and 18 because that will give us the smallest number divisible by both.
  2. Remainders: A remainder is the amount left over after division. We know our number leaves a remainder of 5 when divided by 12 and 18. This means our number is 5 more than a multiple of both 12 and 18.

Finding the Solution

Now, let's dive into the steps to find our mystery number.

Step 1: Find the Least Common Multiple (LCM) of 12 and 18

There are a couple of ways to find the LCM. One common method is prime factorization:

  • Prime factorization of 12: 2 x 2 x 3 (or 2² x 3)
  • Prime factorization of 18: 2 x 3 x 3 (or 2 x 3²)

To find the LCM, we take the highest power of each prime factor present in either number:

  • Highest power of 2: 2²
  • Highest power of 3: 3²

So, the LCM of 12 and 18 is 2² x 3² = 4 x 9 = 36. This means 36 is the smallest number divisible by both 12 and 18.

Step 2: Find Multiples of the LCM

We know that our target number, minus 5, must be a multiple of 36. So, let's list out some multiples of 36:

  • 36 x 1 = 36
  • 36 x 2 = 72
  • 36 x 3 = 108
  • 36 x 4 = 144
  • And so on...

Step 3: Add the Remainder

Remember, our target number leaves a remainder of 5 when divided by 12 and 18. This means we need to add 5 to each of the multiples of 36 we just found:

  • 36 + 5 = 41
  • 72 + 5 = 77
  • 108 + 5 = 113
  • 144 + 5 = 149
  • And so on...

Step 4: Identify the Smallest 3-Digit Number

Now, we look at the numbers we generated by adding 5 and find the smallest one that is a three-digit number. Looking at our list, we see that 113 is the smallest three-digit number that fits our criteria!

Therefore, the Answer is 113

So, the smallest three-digit natural number that leaves a remainder of 5 when divided by both 12 and 18 is 113. Isn't math fun when you break it down like this? We used the concepts of LCM and remainders to solve this problem, and you can apply these same principles to other similar problems too.

Why This Works: A Deeper Dive

Let's quickly recap why this method works. By finding the LCM of 12 and 18, we found the smallest number that both 12 and 18 divide into evenly. When we added the remainder (5) to multiples of the LCM, we ensured that our resulting number would always leave a remainder of 5 when divided by 12 and 18.

We then simply searched through these numbers until we found the smallest one that also satisfied the three-digit condition. This approach guarantees that we find the smallest three-digit number that meets all the requirements of the problem.

Alternative Approach: Trial and Error (and Why It's Less Efficient)

You could try to solve this problem using trial and error, but it would be much less efficient. You might start by checking numbers like 100, 101, 102, and so on, dividing each by 12 and 18 to see if the remainder is 5. However, this would take a lot of time and there's no guarantee you'd find the smallest solution quickly.

The LCM method is much more systematic and ensures you find the correct answer efficiently.

Practice Makes Perfect

The best way to get comfortable with these types of problems is to practice! Try these variations:

  • What is the smallest three-digit number that leaves a remainder of 3 when divided by 8 and 10?
  • What is the smallest four-digit number that leaves a remainder of 2 when divided by 9 and 15?

By working through similar problems, you'll strengthen your understanding of LCM, remainders, and problem-solving strategies in general.

Conclusion

We successfully found the smallest three-digit number that leaves a remainder of 5 when divided by 12 and 18. The answer is 113. Remember the key steps: find the LCM, add the remainder to multiples of the LCM, and then identify the smallest number that meets all conditions. Keep practicing, and you'll become a math whiz in no time!

If you have any questions or want to explore more math problems, feel free to ask. Happy problem-solving, guys!

Understanding Remainders in More Detail

Let's dive a little deeper into the concept of remainders, as it's fundamental to solving this type of problem. When we divide one number (the dividend) by another (the divisor), the remainder is what's left over if the divisor doesn't divide the dividend evenly.

For example, if we divide 29 by 7, we get 4 with a remainder of 1. This is because 7 goes into 29 four times (7 x 4 = 28), and there's 1 left over (29 - 28 = 1).

In our original problem, the fact that we have a remainder of 5 when dividing by both 12 and 18 tells us that the number we're looking for is 5 more than a number that's perfectly divisible by both 12 and 18. This is why we subtracted 5 initially and focused on finding multiples of the LCM.

Think of it like this: if we had a pile of candies and wanted to divide it equally among 12 people and also among 18 people, and we always had 5 candies left over, then the total number of candies must be 5 more than a number that's divisible by both 12 and 18.

The Significance of the Least Common Multiple (LCM)

The LCM plays a crucial role in this problem, and in many other number theory problems. As we discussed earlier, the LCM of two numbers is the smallest number that is a multiple of both. Why is this important?

In our case, we need a number that, when we subtract 5, is divisible by both 12 and 18. Using the LCM ensures that we find the smallest such number. If we used any common multiple other than the LCM, we would end up with a larger solution, and we're specifically looking for the smallest three-digit number.

For example, while 36 is the LCM of 12 and 18, other common multiples include 72, 108, 144, and so on. If we used 72 instead of 36, we would eventually find a solution, but it would be larger than 113. Using the LCM streamlines the process and guarantees we find the minimal solution.

Generalizing the Problem-Solving Strategy

The strategy we used to solve this problem can be generalized to a wider range of similar questions. Here's a breakdown of the general approach:

  1. Understand the problem: Carefully identify what you're looking for (e.g., the smallest number, a number within a specific range) and the given conditions (e.g., remainders after division).
  2. Identify key concepts: Determine which mathematical concepts are relevant (e.g., LCM, Greatest Common Divisor (GCD), remainders, divisibility rules).
  3. Formulate a plan: Outline the steps you'll take to solve the problem. This might involve finding the LCM or GCD, manipulating equations, or using specific formulas.
  4. Execute the plan: Carry out the steps you outlined, showing your work clearly.
  5. Check your answer: Make sure your answer satisfies all the conditions of the problem and is reasonable in the context of the question.

By following this general problem-solving strategy, you can tackle a wide variety of math challenges with confidence.

Real-World Applications of LCM and Remainders

While these concepts might seem purely theoretical, they actually have practical applications in everyday life. Here are a couple of examples:

  • Scheduling: Imagine you have two tasks to complete. One task needs to be done every 12 days, and the other needs to be done every 18 days. The LCM of 12 and 18 (which is 36) tells you how often both tasks will need to be done on the same day. This helps with scheduling and planning.
  • Dividing Items: If you have a certain number of items and want to divide them equally among a group of people, the remainder tells you how many items will be left over. This is useful in situations like sharing snacks or distributing resources.

Understanding LCM and remainders can help you solve real-world problems more efficiently and make informed decisions.

Further Exploration

If you're interested in learning more about number theory and related concepts, there are many resources available online and in libraries. You can explore topics like:

  • Prime numbers: Numbers that are only divisible by 1 and themselves.
  • Greatest Common Divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
  • Modular arithmetic: A system of arithmetic for integers where numbers "wrap around" upon reaching a certain value (the modulus).
  • Diophantine equations: Equations where the solutions must be integers.

By continuing to explore these topics, you'll deepen your understanding of mathematics and develop valuable problem-solving skills. Keep up the great work, everyone!