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Chapter 3: Problem 19

Find the LCM of each set of numbers. $$ 32,36 $$

Short Answer

Expert verified
The LCM is 288.

Step by step solution

01

Prime Factorization of 32

To find the LCM, first perform the prime factorization of 32. The prime factors of 32 are: 32 = 2 脳 2 脳 2 脳 2 脳 2 = 2^5
02

Prime Factorization of 36

Next, perform the prime factorization of 36. The prime factors of 36 are: 36 = 2 脳 2 脳 3 脳 3 = 2^2 脳 3^2
03

Identify the Highest Powers of All Prime Factors

The prime factors we have are 2 and 3. Take the highest power of each:Highest power of 2: 2^5Highest power of 3: 3^2
04

Calculate the LCM

Multiply the highest powers of all prime factors:LCM = 2^5 脳 3^2Calculate the product:2^5 = 323^2 = 9LCM = 32 脳 9 = 288

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Key Concepts

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

Prime Factorization
Prime factorization involves breaking down a number into its basic building blocks 鈥� the prime numbers. Let's take a closer look at how we performed prime factorization for the numbers 32 and 36 from the exercise.

For 32:
  • 32 can be factored into 2 脳 2 脳 2 脳 2 脳 2, which can be written as 25.
For 36:
  • 36 can be factored into 2 脳 2 脳 3 脳 3, which can be written as 22 脳 32.
Remember, prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The main prime numbers we used here are 2 and 3.

Understanding this concept is essential for calculating the Least Common Multiple (LCM).
Highest Powers
When calculating the LCM, we need to use the highest powers of all prime factors present in the numbers. Using the same example, let's see how we determine the highest powers.

First, list all the prime factors and then take note of their highest powers:
  • For the number 32, the highest power of 2 is 25.
  • For the number 36, the highest power of 3 is 32.
Therefore, when looking at both numbers together, we take:
  • The highest power of 2 is 25.
  • The highest power of 3 is 32.
This step ensures that the LCM contains all necessary prime factors in their most significant forms.
LCM Calculation
Now that we have the highest powers of all prime factors, we can calculate the LCM by multiplying them together. From our example:

The highest power of 2 is 25.
The highest power of 3 is 32.

So the LCM is: \[ \text{LCM} = 2^{5} \times 3^{2} = 32 \times 9 \] \[ \text{LCM} = 288 \]

By multiplying these, you incorporate all prime factors at their highest powers, giving you the smallest number that both original numbers can divide into without any remainder.
Understanding this systematic approach simplifies finding the LCM of any set of numbers.

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Most popular questions from this chapter

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