Finding The Least Number To Subtract For Divisibility By 54 And 72

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Introduction

In the realm of mathematics, number theory presents fascinating challenges that require a blend of logical reasoning and arithmetic skills. One such challenge involves finding the smallest number that, when subtracted from a given number, results in a new number perfectly divisible by two or more specified divisors. This type of problem not only tests our understanding of divisibility rules but also sharpens our ability to apply concepts like the Least Common Multiple (LCM) and remainders. In this article, we will delve into a step-by-step solution to the problem of finding the least number to subtract from 796976 to make it exactly divisible by both 54 and 72. By understanding the underlying principles and the step-by-step approach, readers will be well-equipped to tackle similar problems with confidence. The core objective here is to ensure that after subtracting the least possible number, the resultant value is a multiple of both 54 and 72. This involves finding a common multiple and then identifying the smallest subtraction needed to achieve this divisibility. This mathematical puzzle not only enhances our problem-solving skills but also deepens our understanding of number relationships and divisibility rules. So, let’s embark on this mathematical journey and unravel the solution together!

Understanding the Problem

Before diving into the solution, it's crucial to fully grasp the problem statement. We are given the number 796976 and two divisors, 54 and 72. Our mission is to determine the smallest number that can be subtracted from 796976 such that the resulting number is perfectly divisible by both 54 and 72. This means that when the new number is divided by 54 and 72, the remainder should be zero. To effectively tackle this problem, we need to understand the concepts of divisibility and the Least Common Multiple (LCM). Divisibility refers to the ability of one number to be divided by another without leaving a remainder. For instance, 12 is divisible by 3 because 12 ÷ 3 = 4 with no remainder. The Least Common Multiple (LCM), on the other hand, is the smallest positive integer that is divisible by both numbers. For example, the LCM of 6 and 8 is 24 because 24 is the smallest number that both 6 and 8 divide into evenly. The significance of LCM in this problem is that it gives us a target number that is a multiple of both divisors. Once we find the LCM of 54 and 72, we can then work towards finding a multiple of this LCM that is closest to 796976 but less than it. This will help us determine the number to be subtracted. Understanding these foundational concepts is key to unlocking the solution. In the following sections, we will explore how to calculate the LCM and then apply it to solve the problem at hand.

Step 1: Finding the Least Common Multiple (LCM) of 54 and 72

To solve the problem efficiently, the first crucial step is to determine the Least Common Multiple (LCM) of the divisors, which are 54 and 72 in this case. The LCM is the smallest positive integer that is divisible by both numbers. There are several methods to find the LCM, but one of the most common and straightforward approaches is the prime factorization method. This method involves breaking down each number into its prime factors and then using those factors to compute the LCM. Let's start by finding the prime factorization of 54 and 72. The prime factorization of 54 is 2 × 3 × 3 × 3, which can be written as 2 × 3³. Similarly, the prime factorization of 72 is 2 × 2 × 2 × 3 × 3, which can be expressed as 2³ × 3². Now that we have the prime factorizations, we can calculate the LCM. To do this, we identify the highest power of each prime factor that appears in either factorization. For the prime factor 2, the highest power is 2³ (from the factorization of 72). For the prime factor 3, the highest power is 3³ (from the factorization of 54). To find the LCM, we multiply these highest powers together: LCM(54, 72) = 2³ × 3³ = 8 × 27 = 216. Therefore, the LCM of 54 and 72 is 216. This means that 216 is the smallest number that is divisible by both 54 and 72. Finding the LCM is a critical step because it provides the basis for identifying the target number that we want 796976 to become after subtraction. In the next step, we will use this LCM to find a multiple of 216 that is close to 796976.

Step 2: Dividing 796976 by the LCM (216)

Now that we have determined the Least Common Multiple (LCM) of 54 and 72 to be 216, the next step is to find a multiple of 216 that is close to 796976. This will help us identify the number we need to subtract from 796976 to make it perfectly divisible by both 54 and 72. To find this multiple, we divide 796976 by the LCM, which is 216. This division will give us a quotient, which indicates how many times 216 goes into 796976, and a remainder, which represents the excess amount that makes 796976 not perfectly divisible by 216. Performing the division, we have: 796976 ÷ 216 ≈ 3690. We get a quotient of approximately 3690 and a remainder. The exact calculation is as follows:

 796976 ÷ 216 = 3690 with a remainder of 136

This means that 796976 is 3690 times 216, with an additional 136. In other words, 796976 can be expressed as: 796976 = (3690 × 216) + 136. The remainder of 136 is crucial because it represents the amount by which 796976 exceeds a multiple of 216. This remainder is the key to finding the least number we need to subtract. If we subtract this remainder from 796976, the resulting number will be a perfect multiple of 216, and therefore divisible by both 54 and 72. In the next step, we will use this information to determine the least number to subtract from 796976.

Step 3: Determining the Least Number to Subtract

Having divided 796976 by the LCM of 54 and 72, which is 216, we found that the remainder is 136. This remainder holds the key to solving our problem. As we established in the previous step, the remainder represents the amount by which 796976 exceeds a multiple of 216. Therefore, if we subtract this remainder from 796976, we will obtain a number that is perfectly divisible by 216, and consequently, by both 54 and 72. Thus, the least number to subtract from 796976 to make it exactly divisible by 54 and 72 is the remainder we found, which is 136. To verify this, we can subtract 136 from 796976 and check if the result is divisible by 216: 796976 - 136 = 796840. Now, let's divide 796840 by 216: 796840 ÷ 216 = 3690. Since the division results in a whole number (3690) with no remainder, it confirms that 796840 is indeed divisible by 216. This also means that 796840 is divisible by both 54 and 72, as 216 is their LCM. Therefore, the least number that needs to be subtracted from 796976 to make it exactly divisible by 54 and 72 is 136. This completes our solution. In summary, we found the LCM of the divisors, divided the given number by the LCM to find the remainder, and then identified the remainder as the number to be subtracted. This methodical approach ensures that we arrive at the correct answer efficiently. In the concluding section, we will summarize the entire process and highlight the key concepts used in solving the problem.

Conclusion

In this detailed exploration, we successfully navigated through the problem of finding the least number to subtract from 796976 to make it exactly divisible by 54 and 72. By breaking down the problem into manageable steps, we were able to apply mathematical concepts effectively and arrive at the correct solution. Let's recap the key steps we undertook:

  1. Understanding the Problem: We began by thoroughly understanding the problem statement, which involved identifying the smallest number to subtract from 796976 to ensure divisibility by both 54 and 72.
  2. Finding the LCM: We recognized the importance of the Least Common Multiple (LCM) and calculated the LCM of 54 and 72 using the prime factorization method. We found that LCM(54, 72) = 216.
  3. Dividing by the LCM: We divided 796976 by the LCM (216) to determine the quotient and remainder. The division yielded a remainder of 136.
  4. Determining the Least Number: We identified the remainder (136) as the least number to subtract from 796976 to achieve divisibility by 54 and 72. We verified this by subtracting 136 from 796976 and confirming that the result (796840) is divisible by 216.

This problem-solving exercise highlights the significance of understanding fundamental mathematical concepts such as divisibility and LCM. It also underscores the importance of a systematic approach to problem-solving. By following a step-by-step method, we can break down complex problems into simpler, more manageable parts, making the solution process more accessible and less daunting. The key takeaway from this exercise is the application of LCM in solving divisibility problems. The LCM provides a common multiple that simplifies the process of finding a number divisible by multiple divisors. This approach can be applied to a wide range of similar problems, making it a valuable tool in mathematical problem-solving. Through this detailed explanation, we hope to have provided a clear and comprehensive understanding of how to tackle such problems effectively. The skill of problem-solving in mathematics is not just about arriving at the correct answer, but also about understanding the process and the underlying principles. This understanding enables us to apply our knowledge to new and varied challenges, fostering a deeper appreciation for the beauty and logic of mathematics.