Problem 47 - Algebra A Combined Function

Question

Objective C Find the LCM of each list of numbers. See Examples 7 through 9 . 40,90

Step-by-Step Solution

Answer

The LCM of 40 and 90 is 360.

Step 1: Prime Factorization of 40

First, we need to find the prime factors of 40. Divide 40 by the smallest prime number until we reach 1. \[ 40 \div 2 = 20 \] \[ 20 \div 2 = 10 \] \[ 10 \div 2 = 5 \] \[ 5 \div 5 = 1 \] Therefore, the prime factorization of 40 is \( 2^3 \times 5 \).

Step 2: Prime Factorization of 90

Next, find the prime factors of 90 using the same method.\[ 90 \div 2 = 45 \]\[ 45 \div 3 = 15 \]\[ 15 \div 3 = 5 \]\[ 5 \div 5 = 1 \]Therefore, the prime factorization of 90 is \( 2 \times 3^2 \times 5 \).

Step 3: Identify the Highest Powers of Each Prime

Now, identify the highest powers of each prime number from the factorizations. For 2: The highest power is \(2^3\).For 3: The highest power is \(3^2\).For 5: The highest power is \(5^1\).

Step 4: Calculate the LCM using the Highest Powers

The LCM is found by multiplying together the highest powers of all prime factors:\[ LCM = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 \] Calculate:\[ 8 \times 9 = 72 \]\[ 72 \times 5 = 360 \]

Step 5: Finalize the LCM

Conclude by stating that the least common multiple (LCM) of 40 and 90 is 360.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Common examples include 2, 3, 5, 7, and 11. Every integer greater than 1 can be represented as a product of primes, which is what we call prime factorization.
For instance, to find the prime factorization of 40, you start by dividing the number by the smallest prime number, repeatedly, until you end up with 1. In this case, 40 can be divided by 2 (the smallest prime) to give 20, then continue dividing by 2 until reaching 5 (also a prime number) which gives us the factorization of 40 as:

• 40 ÷ 2 = 20
• 20 ÷ 2 = 10
• 10 ÷ 2 = 5
• 5 ÷ 5 = 1

This results in the prime factors of 40 being represented as \(2^3 \times 5\). Understanding this is crucial for finding the LCM as it forms the basis for comparing the prime factors of different numbers.

Highest Power of Primes

When calculating the LCM of two or more numbers, the highest powers of primes are central to ensuring it is the smallest common multiple. Once you have the prime factorization of each number, you will identify the greatest exponent (or power) for each prime number present across all numbers. This is what we refer to as the "highest power of primes".
From our prime factorizations earlier, we saw:

• 40 is \(2^3 \times 5\)
• 90 is \(2 \times 3^2 \times 5\)

To calculate the LCM, observe each prime and choose the highest power among the numbers. In this case:

• For the prime 2, the highest power is \(2^3\).
• For the prime 3, the highest power is \(3^2\).
• For the prime 5, the highest power is \(5^1\).

Engaging with this step ensures the LCM accounts for all primes at their largest presence in any of the numbers.

Calculating LCM

The Least Common Multiple (LCM) is a critical concept when you need a common ground for different numbers. It's the smallest number that is a multiple of each number in a set. To find it using prime factorization, you multiply the highest powers of each prime number obtained from all the numbers involved.
From our previous analyses:

• The highest power of 2 was \(2^3\).
• The highest power of 3 was \(3^2\).
• The highest power of 5 was \(5^1\).

Thus, the LCM can be calculated by multiplying these together:\[LCM = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5\]Carrying out the multiplications step by step:

• 8 times 9 equals 72
• 72 times 5 equals 360

Hence, the LCM of 40 and 90 is 360. By understanding and applying these concepts, you can solve similar problems with ease.