Math Problem Statement

Find the lcm

Find the LCM of the given pairs using the division method: (12, 18), (36, 24), (18, 15), (6, 14), (30, 16).

Solution

To find the Least Common Multiple (LCM) using the division method for the given pairs of numbers, you can follow these steps:

Steps for Division Method to Find LCM:

  1. Write the numbers side by side.
  2. Divide by the smallest prime number that can divide at least one of the numbers.
  3. Write the quotient below and repeat the process until you cannot divide further (i.e., until all numbers are 1).

Now, let's find the LCM for the pairs in your list:

  1. 12, 18
  2. 36, 24
  3. 18, 15
  4. 6, 14
  5. 30, 16

1. LCM of 12 and 18:

2 & 12 & 18 \\ 2 & 6 & 9 \\ 3 & 3 & 9 \\ & 1 & 3 \\ 3 & 1 & 1 \\ \end{array}$$ LCM = $$2 \times 2 \times 3 \times 3 = 36$$ ### 2. LCM of 36 and 24: $$\begin{array}{c|cc} 2 & 36 & 24 \\ 2 & 18 & 12 \\ 3 & 9 & 6 \\ 3 & 3 & 2 \\ 2 & 1 & 2 \\ 2 & 1 & 1 \\ \end{array}$$ LCM = $$2 \times 2 \times 3 \times 3 \times 2 = 72$$ ### 3. LCM of 18 and 15: $$\begin{array}{c|cc} 3 & 18 & 15 \\ 3 & 6 & 5 \\ 2 & 2 & 5 \\ & 1 & 5 \\ 5 & 1 & 1 \\ \end{array}$$ LCM = $$3 \times 3 \times 2 \times 5 = 90$$ ### 4. LCM of 6 and 14: $$\begin{array}{c|cc} 2 & 6 & 14 \\ 3 & 3 & 7 \\ & 1 & 7 \\ 7 & 1 & 1 \\ \end{array}$$ LCM = $$2 \times 3 \times 7 = 42$$ ### 5. LCM of 30 and 16: $$\begin{array}{c|cc} 2 & 30 & 16 \\ 2 & 15 & 8 \\ 2 & 15 & 4 \\ 2 & 15 & 2 \\ 3 & 5 & 1 \\ 5 & 1 & 1 \\ \end{array}$$ LCM = $$2 \times 2 \times 2 \times 2 \times 3 \times 5 = 240$$ --- Here are the LCMs for each pair: 1. LCM(12, 18) = **36** 2. LCM(36, 24) = **72** 3. LCM(18, 15) = **90** 4. LCM(6, 14) = **42** 5. LCM(30, 16) = **240** Do you want more details or have any questions? ### Related Questions: 1. How do you determine the LCM of more than two numbers using the division method? 2. What is the difference between the division method and the prime factorization method for finding LCM? 3. Can the division method be used to find the greatest common divisor (GCD) as well? 4. How does the LCM of a pair of numbers relate to their product and GCD? 5. How can you verify that your calculated LCM is correct? ### Tip: When dividing by primes, always start with the smallest prime number (2) and move upwards. This minimizes steps and reduces errors.

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Math Problem Analysis

Mathematical Concepts

Least Common Multiple (LCM)
Division Method
Prime Factorization

Formulas

LCM = Product of primes used in division
LCM of two numbers = (Product of numbers) / GCD

Theorems

Prime Factorization
Division Method for LCM

Suitable Grade Level

Grades 6-8

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