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Chapter 3: Problem 44

Find the LCM of each set of numbers. $$ 75,100 $$

Short Answer

Expert verified
The LCM is 300.

Step by step solution

01

Prime Factorization

Find the prime factorization of each number. For 75, we have: \[ 75 = 3 \times 5^2 \] For 100, we have: \[ 100 = 2^2 \times 5^2 \]
02

Identify the Highest Powers of Each Prime

List all primes appearing in the factorizations and identify the highest power of each prime. The primes are 2, 3, and 5. The highest powers are: \[ 2^2, 3^1, 5^2 \]
03

Multiply the Highest Powers Together

The least common multiple (LCM) is found by multiplying the highest powers of all the primes together: \[ LCM = 2^2 \times 3^1 \times 5^2 \] Calculate the product: \[ LCM = 4 \times 3 \times 25 = 300 \]

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Key Concepts

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

Prime Factorization
When solving for the least common multiple (LCM), the first crucial step is prime factorization. Prime factorization means breaking down a number into the set of prime numbers that, when multiplied together, give you the original number. For example:
鈥� For 75, the prime factorization is: \[ 75 = 3 \times 5^2 \] Here, 3 and 5 are prime numbers.
鈥� For 100, the prime factorization is: \[ 100 = 2^2 \times 5^2 \] Here, 2 and 5 are prime numbers.
Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. Prime factorization simplifies finding the highest powers of primes, which is essential for calculating the LCM.
Least Common Multiple
The least common multiple (LCM) is the smallest multiple that is evenly divisible by each number in a set of numbers. To find the LCM using prime factorization, first determine the highest powers of the prime factors involved.
From the prime factorizations:
鈥� For 75: \[ 3^1 \times 5^2 \]
鈥� For 100: \[ 2^2 \times 5^2 \]
We then take the highest power of each prime factor:
鈥� For 2: \[ 2^2 \]
鈥� For 3: \[ 3^1 \]
鈥� For 5: \[ 5^2 \]
Finally, multiply these together to find the LCM:
\[ LCM = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300 \] The LCM of 75 and 100 is 300, the smallest number that both can evenly divide.
Highest Powers of Primes
To find the LCM using prime factorization, identify the highest powers of each prime factor. This step ensures that all components of both numbers are included in the LCM. Here鈥檚 why each prime is chosen:
鈥� **Prime 2:** Appears in 100 as \[ 2^2 \] and not in 75. We take \[ 2^2 \] since it's the highest power.
鈥� **Prime 3:** Appears in 75 as \[ 3^1 \], so \[ 3^1 \] is used.
鈥� **Prime 5:** Appears in both 75 and 100 as \[ 5^2 \]. Hence, \[ 5^2 \] is used.
This method ensures the LCM is the smallest number that contains all prime factors at their highest necessary powers. Always verify your factors by multiplying them to check their compatibility with the given numbers.

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Most popular questions from this chapter

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