Problem 54 - Algebra A Combined Function

Question

Objective C Find the LCM of each list of numbers. See Examples 7 through 9 . 3,5,7

Step-by-Step Solution

Answer

The LCM of 3, 5, and 7 is 105.

Step 1: Understand the Problem

We need to find the Least Common Multiple (LCM) of the numbers 3, 5, and 7. LCM is the smallest number that is a multiple of each of the given numbers.

Step 2: List the Multiples of Each Number

Start by listing the first few positive multiples of each number: - Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... - Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ... - Multiples of 7: 7, 14, 21, 28, 35, 42, ... This will help us identify a common multiple.

Step 3: Identify the Common Multiples

Look for the smallest number that appears in each list of multiples. In this case: - The multiples of 3, 5, and 7 do not have a common number within the first few listed multiples. We need to identify a common number.

Step 4: Calculate the LCM Using Prime Factorization

Another efficient method is prime factorization:- The number 3 is a prime number, expressed as 3.- The number 5 is a prime number, expressed as 5.- The number 7 is a prime number, expressed as 7.Since all numbers are primes, each must be included in the LCM. Therefore, multiply them together: \[ ext{LCM} = 3 imes 5 imes 7 = 105 \]

Step 5: Verify the LCM

Ensure that 105 is divisible by each of the numbers 3, 5, and 7. - 105 ÷ 3 = 35 - 105 ÷ 5 = 21 - 105 ÷ 7 = 15 Since 105 is divisible by all three numbers, it confirms that 105 is indeed the LCM.

Key Concepts

Multiples

When it comes to understanding Least Common Multiple (LCM), it's crucial to get to grips with what multiples are. Multiples of a number are the products you get when multiplying the number by integers. For example, if we take the number 3, its multiples are 3, 6, 9, 12, and so forth. These are all numbers you get by multiplying 3 by 1, 2, 3, 4, respectively.

It's like creating a sequence based on continuous addition of the number itself.

• Multiples of 3: 3, 6, 9, 12, 15, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...
• Multiples of 7: 7, 14, 21, 28, 35, ...

Understanding multiples is the first step towards finding the LCM, as LCM is essentially the smallest shared multiple among a set of numbers. This concept helps to deeply comprehend how different numbers interact in terms of their product sequences.

Prime Factorization

Prime factorization is another effective method to find the Least Common Multiple. This method breaks down each number into a product of prime numbers. Prime numbers themselves have no divisors other than 1 and the number itself. So, when you express a number as a product of its prime factors, you're laying out its basic building blocks.

For instance:

• The number 3 is a prime, hence it is expressed as 3.
• The number 5 is also a prime, expressed as 5.
• Finally, the number 7 is a prime too, written as 7.

Since each of these numbers is prime, they should all be included in finding the LCM. To do so, multiply them together: \[\text{LCM} = 3 \times 5 \times 7 = 105\]This way, the product gives you the smallest number that all original numbers divide into without leaving a remainder. Prime factorization simplifies finding LCM, especially when dealing with larger numbers.

Divisibility

Understanding divisibility is key to verifying that you've correctly identified the Least Common Multiple. A number is divisible by another if you can divide it exactly without leaving a remainder. For example, to check if 105 is the LCM of 3, 5, and 7, we ensure 105 is divisible by each of these numbers.

Let's verify this:

• 105 divided by 3: 105 ÷ 3 = 35, which is an integer.
• 105 divided by 5: 105 ÷ 5 = 21, which is an integer.
• 105 divided by 7: 105 ÷ 7 = 15, which is an integer.

If a number is divisible by each number in your list, and it's the smallest such number, congrats! You've found your LCM. Divisibility helps ensure accuracy when identifying the correct LCM, providing a reliable way to double-check your work.