L.C.M with Roman Numerals

Introduction

The Least Common Multiple (L.C.M) is the smallest multiple that two or more numbers share. In this lesson, we will explore how to find the L.C.M using Roman numerals, an ancient numeral system. Understanding L.C.M is essential in solving various mathematical problems, especially in fractions and ratios.

Word Problem

Calculating the L.C.M

Step 1: Convert Roman Numerals to Arabic Numerals

The first step is to convert the Roman numerals into Arabic numerals:

• IX = 9
• XII = 12

Step 2: Prime Factorization

Next, we will find the prime factorization of each number:

• 9 = 3 × 3 = $3^2$
• 12 = 3 × 2 × 2 = $2^2 \cdot 3^1$

Step 3: Determine the L.C.M

The L.C.M can be calculated by taking the highest power of each prime factor that appears in the factorizations:

Using the formula:

$$\text{L.C.M}(a, b) = \prod_{i} p_i^{\text{max}(e_{ai}, e_{bi})}$$

In our case:

$$\text{L.C.M}(9, 12) = 2^2 \times 3^2 = 4 \times 9 = 36$$

Conclusion

The least common multiple of IX (9) and XII (12) is XXXVI (36). Understanding the process of finding the L.C.M is important, as it helps in solving problems involving fractions and finding common denominators.