Lowest Common Multiple Of 36 And 45

Finding the Lowest Common Multiple (LCM) of 36 and 45

This article will guide you through calculating the lowest common multiple (LCM) of 36 and 45. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cyclical events. We'll explore two common methods: prime factorization and the least common multiple formula.

What is the Lowest Common Multiple (LCM)?

The lowest common multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. In simpler terms, it's the smallest number that both 36 and 45 can divide into without leaving a remainder. This concept is fundamental in algebra, number theory, and even everyday problem-solving.

Method 1: Prime Factorization

This method involves breaking down each number into its prime factors. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

1. Find the prime factorization of 36: 36 = 2 x 2 x 3 x 3 = 2² x 3²

2. Find the prime factorization of 45: 45 = 3 x 3 x 5 = 3² x 5

3. Identify the highest power of each prime factor present in either factorization: The prime factors are 2, 3, and 5. The highest power of 2 is 2², the highest power of 3 is 3², and the highest power of 5 is 5.

4. Multiply the highest powers together: LCM(36, 45) = 2² x 3² x 5 = 4 x 9 x 5 = 180

Therefore, the lowest common multiple of 36 and 45 is 180. This means 180 is the smallest number that is divisible by both 36 and 45.

Method 2: Using the Formula (LCM and GCD)

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

LCM(a, b) = (|a x b|) / GCD(a, b)

where:

• a and b are the two numbers (36 and 45 in our case)
• GCD(a, b) is the greatest common divisor of a and b.

1. Find the GCD of 36 and 45: We can use the Euclidean algorithm to find the GCD.

   • 45 = 1 x 36 + 9
   • 36 = 4 x 9 + 0 The last non-zero remainder is 9, so GCD(36, 45) = 9.

2. Apply the formula: LCM(36, 45) = (36 x 45) / 9 = 1620 / 9 = 180

Again, we find that the LCM of 36 and 45 is 180.

Conclusion:

Both methods effectively calculate the LCM of 36 and 45. The prime factorization method is generally easier to understand for beginners, while the formula method can be more efficient for larger numbers, especially when using a calculator to determine the GCD. Understanding LCM is a fundamental skill in mathematics with various practical applications. Choosing the best method depends on your comfort level and the specific numbers involved. Remember to always double-check your calculations to ensure accuracy.