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Chapter 6: Problem 39

Find the LCM of 32 and 50 .

Short Answer

Expert verified
The LCM of 32 and 50 is 800.

Step by step solution

01

Find the Prime Factors

Start by finding the prime factorization of each number. For 32: 32 = 2^5 For 50: 50 = 2 脳 5^2
02

Identify the Highest Power of Each Prime

List the highest power of each prime number that appears in the factorization of each number. Prime factors: 2, 5 Highest powers found: 2^5 (from 32) 5^2 (from 50)
03

Calculate the LCM

Multiply the highest powers of all the prime factors together. LCM = 2^5 脳 5^2 LCM = 32 脳 25 = 800

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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 smallest prime numbers that, when multiplied together, give the original number. A prime number is only divisible by 1 and itself. For instance, to find the prime factors of 32, you keep dividing by the smallest prime, which is 2, until you can no longer divide evenly.

For 32:
32 梅 2 = 16
16 梅 2 = 8
8 梅 2 = 4
4 梅 2 = 2
2 梅 2 = 1
So, 32 = 2^5

Similarly, let's break down 50:
50 梅 2 = 25
25 梅 5 = 5
5 梅 5 = 1
So, 50 = 2^1 脳 5^2.

Prime factorization is crucial because it lays the foundation for calculating the Least Common Multiple (LCM).
LCM Calculation
The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both. To find the LCM using prime factorization, follow these steps:

  • List out the prime factors of each number.
  • Identify the highest power of each prime factor that appears in the factorization of both numbers.
  • Multiply these highest powers together to get the LCM.

For the given numbers, their prime factorizations are:
32 = 2^5
50 = 2 脳 5^2
The highest power of 2 occurring is 2^5, and the highest power of 5 is 5^2.

So, to find the LCM, multiply these highest powers:
LCM = 2^5 脳 5^2
LCM = 32 脳 25 = 800.

The result, 800, is the smallest common multiple of 32 and 50.
Mathematics Steps
Following the proper steps in mathematics ensures accuracy in your calculations. Here's a more detailed breakdown of the steps used in this exercise:

  • Step 1: Prime Factorization - Determine the prime factors of each number. For 32, it's 2^5, and for 50, it's 2^1 脳 5^2.
  • Step 2: Identify Highest Powers - Look at each prime factor and determine the highest power that appears in any of the factorizations. In this case, for the prime number 2, the highest power is 2^5, and for the prime number 5, the highest power is 5^2.
  • Step 3: Calculate the LCM - Multiply these highest powers together to get the LCM. Therefore, 2^5 脳 5^2 = 32 脳 25 = 800.

Remember each step builds on the previous one, so ensure accuracy at every stage. Taking your time to go through these steps methodically will make sure you find the correct LCM every time.

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