Least Common Multiple Of 30 And 42

Finding the Least Common Multiple (LCM) of 30 and 42

This article will guide you through finding the least common multiple (LCM) of 30 and 42. Understanding LCMs is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and periods. We'll explore two common methods: prime factorization and the least common multiple formula.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all the integers. For instance, the LCM of 2 and 3 is 6, because 6 is the smallest positive number divisible by both 2 and 3. Finding the LCM is a fundamental concept in number theory and has practical applications in various fields.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Let's apply it to find the LCM of 30 and 42:

1. Find the prime factorization of each number:

   • 30 = 2 × 3 × 5
   • 42 = 2 × 3 × 7

2. Identify the highest power of each prime factor present in either factorization:

   • The prime factors are 2, 3, 5, and 7.
   • The highest power of 2 is 2¹ = 2
   • The highest power of 3 is 3¹ = 3
   • The highest power of 5 is 5¹ = 5
   • The highest power of 7 is 7¹ = 7

3. Multiply the highest powers together:

   • LCM(30, 42) = 2 × 3 × 5 × 7 = 210

Therefore, the least common multiple of 30 and 42 is 210.

Method 2: Using the Formula (LCM and GCD)

This method utilizes the relationship between the least common multiple (LCM) and the greatest common divisor (GCD) of two numbers. The formula is:

LCM(a, b) = (|a × b|) / GCD(a, b)

Where:

• a and b are the two numbers.
• GCD(a, b) is the greatest common divisor of a and b.

Let's apply this to 30 and 42:

1. Find the GCD of 30 and 42:

   We can use the Euclidean algorithm or prime factorization to find the GCD. Using prime factorization:

   • 30 = 2 × 3 × 5
   • 42 = 2 × 3 × 7
   • The common factors are 2 and 3.
   • GCD(30, 42) = 2 × 3 = 6

2. Apply the formula:

   • LCM(30, 42) = (30 × 42) / 6 = 1260 / 6 = 210

Again, the least common multiple of 30 and 42 is 210.

Conclusion

Both methods provide the same result: the least common multiple of 30 and 42 is 210. Choosing the best method depends on your preference and the complexity of the numbers involved. Prime factorization is often easier to visualize, while the formula is concise and efficient, especially when dealing with larger numbers where finding the GCD is straightforward. Understanding these methods empowers you to solve a range of mathematical problems involving multiples and divisors.