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Chapter 10: Problem 71

Find the LCM. $$ 48,56,64 $$

Short Answer

Expert verified
The LCM of 48, 56, and 64 is 1344.

Step by step solution

01

- Find Prime Factorization

Identify the prime factors of each number.For 48: \[48 = 2^4 \times 3\]For 56: \[56 = 2^3 \times 7\]For 64: \[64 = 2^6\]
02

- Identify Highest Powers of Prime Factors

Determine the highest power of each prime factor:- The highest power of 2: \[2^6\] - The highest power of 3: \[3^1\] - The highest power of 7: \[7^1\]
03

- Multiply the Highest Powers

Multiply the highest powers of all the prime factors: \[2^6 \times 3^1 \times 7^1 = 64 \times 3 \times 7 = 1344\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
To find the Least Common Multiple (LCM) of multiple numbers, the first step is to identify their prime factors. Prime factorization means breaking down a number into the product of prime numbers. Prime numbers are those that have only two factors: 1 and the number itself. For example, the prime factors of 48 are: \ \[48 = 2^4 \times 3\] \ Here, 2 and 3 are both primes. Likewise, for 56: \ \[56 = 2^3 \times 7\] \ And for 64: \ \[64 = 2^6\] \ Breaking numbers down in this way lets us more easily identify the necessary components for calculating their LCM.
Highest Powers of Prime Factors
Once we have the prime factors of each number, the next step is to determine the highest power of each prime factor. The highest power is simply the greatest exponent that the prime factor is raised to, among all given numbers. For our example of the numbers 48, 56, and 64: \ - The highest power of 2 from the factors is \ \[2^6\] \ (because 64 has 2 raised to the power of 6). \ - The highest power of 3 is \ \[3^1\] \ (from 48). \ - The highest power of 7 is \ \[7^1\] \ (from 56). \ Identifying these highest powers is crucial because they will be used to compute the LCM.
Multiplication of Prime Factors
To find the LCM, we now multiply the highest powers of all prime factors together. Using our example: \ \[2^6 \times 3^1 \times 7^1\] \ Expanding this multiplication: \ \[64 \times 3 \times 7\] \ Calculates to: \ \[64 \times 3 = 192\] \ \[192 \times 7 = 1344\] \ Thus, the LCM of 48, 56, and 64 is \ \[1344\] \ By using the highest powers of each prime factor, we ensure that all original numbers can be divided evenly into the LCM without any remainder.

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