The Number of Pages in a Book: A Step-by-Step Guide Using LCM

Often, when solving word problems, we encounter scenarios that require us to find a specific number within a given range that possesses certain divisibility properties. For instance, if a book has a certain number of pages, and when you count these pages in groups of 5 or 6, you find that there are no extra pages, we need to determine the exact number of pages. This article will guide you through solving such a problem using the least common multiple (LCM).

Understanding the Problem

Suppose a book has a number of pages between 100 and 150. When you count the pages in groups of 5 or 6, there are no extra pages. This means the number of pages must be divisible by both 5 and 6.

Step 1: Identify the Prime Factorizations

To find the LCM of 5 and 6, we first need to identify their prime factorizations:

Prime Factorization of 5

5 51

Prime Factorization of 6

6 21 times; 31

Now, to find the LCM, we take the highest power of each prime that appears in the factorizations:

LCM Calculation

LCM 21 times; 31 times; 51 30

Step 2: List Multiples of the LCM Between 100 and 150

Next, we will list the multiples of 30 that fall between 100 and 150. Let's do this calculation:

Multiples of 30 Between 100 and 150

An easy way to list these is by multiplying 30 by consecutive integers:

30 times; 4 120 30 times; 5 150 (excluded because it exceeds 150)

Therefore, the only multiple of 30 between 100 and 150 is 120.

Step 3: Verify the Solution

To verify that 120 is the correct answer, we can perform a quick check to ensure that 120 is divisible by both 5 and 6:

Verification

120 mod 5 0

120 mod 6 0

Both conditions are satisfied, confirming that 120 is the correct number of pages in the book.

Conclusion

By using the least common multiple, we can easily solve problems that involve finding a specific number within a range with certain divisibility properties. In this case, we found that the book has 120 pages, which is both divisible by 5 and 6 within the given range of 100 to 150.

Additional Tips

Prime Factorization: Always start by identifying the prime factorization of the numbers involved to simplify the LCM calculation. Pattern Recognition: Listing multiples can be a quick way to find the desired number within a specified range. Verification: Always verify your solution to ensure it meets all the problem's requirements.

Understanding and applying this method can help you solve similar math problems effectively. Whether you're a student or a professional, these skills are invaluable in both academic and practical contexts.