Determine the Value of n for LCM 8, 12, and n 720 Using Prime Factorization
In this tutorial, we will explore how to determine the value of n when the Least Common Multiple (LCM) of 8, 12, and n is 720. We will use prime factorization and a step-by-step approach to solve the problem, ensuring that the solution is both robust and easy to follow.
Introduction to Prime Factorization
To solve the problem, we need to understand the concept of prime factorization. Prime factorization is the process of determining the prime numbers that multiply together to give the original number. We will use this method to verify the prime factors of 8, 12, and 720.
Prime Factorization of 8
The prime factorization of 8 is:
8 23
Prime Factorization of 12
The prime factorization of 12 is:
12 22 × 31
Prime Factorization of 720
The prime factorization of 720 is:
720 24 × 32 × 51
Step-by-Step Solution
We need to find the value of n such that the LCM of 8, 12, and n is 720. To achieve this, we will compare the prime factorization of 8, 12, and 720 and determine the minimum power for each prime factor required to make 720 the LCM.
Prime Factors of 8 and 12
The prime factors of 8 are:
23The prime factors of 12 are:
22 31Prime Factors of 720
The prime factors of 720 are:
24 32 51Ensuring 720 as the LCM
To ensure 720 is the LCM, each prime factor must be present in n with a power at least as high as it is in 720:
Step 1: Prime Factor 2
Since 2 is not present in 8 or 12, we must include it in n with a power of at least 4 (the highest power in 720).
8 23
12 22 × 31
n 24
Step 2: Prime Factor 3
Since 3 is not present in either 8 or 12, we must include it in n with a power of at least 2 (the highest power in 720).
8 23
12 22 × 31
n 24 × 32
Step 3: Prime Factor 5
Since 5 is not present in either 8 or 12, we must include it in n with a power of at least 1.
8 23
12 22 × 31
n 24 × 32 × 51
When we multiply n 24 × 32 × 51 by both 8 and 12, we get 720 as the LCM. This confirms that n 720.
Conclusion
The value of n for which the LCM of 8, 12, and n is 720 is indeed 720. By using prime factorization and ensuring that the LCM meets the required prime powers, we have reached a robust and accurate solution.