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Chapter 2: Problem 47

Find the least common multiple of the given numbers. \(5,7,\) and 14

Short Answer

Expert verified
The least common multiple of 5, 7, and 14 is 70.

Step by step solution

01

Identify the Prime Factors

List the prime factors of each number. For 5, it is just 5. For 7, it is just 7. For 14, it is 2 and 7.
02

Write Down the Prime Factors

For the number 5: \[ 5 = 5 \] For the number 7: \[ 7 = 7 \] For the number 14: \[ 14 = 2 \times 7 \]
03

Choose the Highest Powers of All Prime Factors

The prime factors from the numbers are 2, 5, and 7. The highest power of each is: \[ 2^1, 5^1, \text{ and } 7^1 \]
04

Multiply the Highest Powers Together

Find the least common multiple by multiplying the highest powers of the prime factors: \[ \text{LCM} = 2^1 \times 5^1 \times 7^1 = 2 \times 5 \times 7 \]
05

Calculate the Product

Multiply the numbers together to get the result: \[ 2 \times 5 = 10 \] Then, multiply 10 by 7: \[ 10 \times 7 = 70 \]

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Key Concepts

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

Prime Factors
Prime factors are the building blocks of all numbers. They are prime numbers that multiply together to create the original number. For example, the number 14 can be broken down into the prime factors 2 and 7.
To find the prime factors of a number, you divide the number by the smallest prime number (2, 3, 5, 7, etc.) until you get 1.
Here鈥檚 how you find the prime factors of 14:
  • 14 is divisible by 2, giving 14 梅 2 = 7
  • 7 is a prime number, so it cannot be divided further
Putting it together, 14 = 2 脳 7
It鈥檚 essential to understand every number has a unique set of prime factors.
LCM Calculation
The Least Common Multiple (LCM) is the smallest number that can be evenly divided by each of the given numbers. Calculating the LCM helps in solving problems involving multiple quantities. For example, if you need to find the LCM of 5, 7, and 14, follow these steps:
1. Identify the prime factors of each number:
  • 5: Prime factor is 5
  • 7: Prime factor is 7
  • 14: Prime factors are 2 and 7
2. Write down these factors with their highest powers:
  • 2: Highest power is 21
  • 5: Highest power is 51
  • 7: Highest power is 71
The LCM will be the product of these prime factors with their highest powers.
This method ensures that the LCM contains all prime factors needed to cover each original number.
Multiplying Prime Factors
Once you've identified and listed the highest powers of all prime factors, the final step in finding the LCM is multiplication.
For example, to find the LCM of 5, 7, and 14 with prime factors 21, 51, and 71 respectively, you will multiply all the highest power prime factors together:
  • Start with 2 脳 5 = 10
  • Then multiply 10 by 7 to get 70
So, the LCM of 5, 7, and 14 is 70.
This step is crucial and must be done carefully to ensure the correct LCM, which is the least number that can be divided by all of the original numbers without any remainder.

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