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Chapter 7: Problem 41

Find the LCM. $$6,12,15$$

Short Answer

Expert verified
The LCM of 6, 12, and 15 is 60.

Step by step solution

01

- Find the prime factors

First, find the prime factors of each number. 6 = 2 * 3, 12 = 2^2 * 3, 15 = 3 * 5.
02

- Identify the highest power of each prime factor

For all numbers, list the prime factors and take the highest power of each prime factor.2: max(2^1, 2^2, 2^0) = 2^2, 3: max(3^1, 3^1, 3^1) = 3^1, 5: max(5^0, 5^0, 5^1) = 5^1.
03

- Multiply the highest powers of all prime factors

Now, multiply the highest powers of each prime factor to find the LCM.LCM = 2^2 * 3^1 * 5^1 = 4 * 3 * 5.
04

- Calculate the result

Finally, calculate the product.LCM = 4 * 3 = 12, 12 * 5 = 60.

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into the set of prime numbers that multiply together to result in the original number. In the given example, the numbers 6, 12, and 15 are broken down as follows:
  • 6 becomes 2 * 3
  • 12 becomes 2^2 * 3
  • 15 becomes 3 * 5
The prime numbers identified are the building blocks of the original numbers. This is crucial because these prime factors will be used to find the Lowest Common Multiple (LCM). Take time to practice getting prime factors; it makes understanding advanced concepts easier.
For clarity, remember: A prime number is a number greater than 1 having no divisors other than 1 and itself.
Highest Power of Prime Factors
Once you've identified the prime factors, the next step is to find the highest power of each prime factor across all numbers. This means you check the list of prime factors and note the maximum exponent or power for each prime.
Let's illustrate:
  • For the prime number 2, we have: 2^1 (from 6), 2^2 (from 12), and 2^0 (from 15). The highest power here is 2^2.
  • For the prime number 3: 3^1 (from 6), 3^1 (from 12), and 3^1 (from 15). The highest power is 3^1.
  • For the prime number 5: 5^0 (from 6), 5^0 (from 12), and 5^1 (from 15). The highest power is 5^1.
It's essential to list these powers one by one for each unique prime factor found, as this will guide the multiplication process to determine the LCM.
Multiplication of Prime Factors
Once we've collected the highest powers of each prime factor, the next step is to multiply them together. This combination will yield the LCM.
For our specific example, the highest powers are:
  • 2^2
  • 3^1
  • 5^1
Now, multiply these highest powers together: \(2^2 \times 3^1 \times 5^1\).
Breaking it down: \(2^2 = 4\), \(3^1 = 3\), and \(5^1 = 5\). First, multiply the 4 and 3 to get 12, then multiply 12 by 5 to get 60.
Hence, the LCM of 6, 12, and 15 is 60.
Practicing these steps of multiplying correctly ensures that you get the right LCM every time. Taking it one step at a time makes it easy and manageable!

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