LCM Calculations: True Or False? Explained!
Hey guys! Let's dive into some Least Common Multiple (LCM) calculations and figure out whether these statements are true or false. We'll break down each one step-by-step, so you can follow along and understand exactly how to calculate the LCM. Get ready to put on your math hats!
Understanding the Least Common Multiple (LCM)
Before we jump into the problems, let's quickly recap what the LCM actually is. The LCM of a set of numbers is the smallest positive integer that is divisible by each of those numbers. Think of it as the smallest number that all the given numbers can fit into evenly. Finding the LCM is super useful in many areas of math, like when you're adding fractions with different denominators.
There are a couple of ways to calculate the LCM, but we're going to focus on the prime factorization method here. This method is reliable and helps you understand the building blocks of each number.
The Prime Factorization Method
Here's the basic idea behind the prime factorization method:
- Find the prime factorization of each number. This means breaking each number down into its prime factors (prime numbers that multiply together to give you the original number). Remember, prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).
- Identify the highest power of each prime factor that appears in any of the factorizations.
- Multiply all those highest powers together. The result is the LCM!
Okay, enough theory! Let's put this into practice with our problems.
a) LCM(270, 405, 162) = 810
Let's start with the first statement: LCM(270, 405, 162) = 810. To determine if this is true, we need to calculate the LCM of 270, 405, and 162 using the prime factorization method.
Step 1: Prime Factorization
First, we'll find the prime factorization of each number:
- 270 = 2 × 135 = 2 × 3 × 45 = 2 × 3 × 3 × 15 = 2 × 3 × 3 × 3 × 5 = 2 × 3³ × 5
- 405 = 5 × 81 = 5 × 3 × 27 = 5 × 3 × 3 × 9 = 5 × 3 × 3 × 3 × 3 = 3⁴ × 5
- 162 = 2 × 81 = 2 × 3 × 27 = 2 × 3 × 3 × 9 = 2 × 3 × 3 × 3 × 3 = 2 × 3⁴
So, we have:
- 270 = 2¹ × 3³ × 5¹
- 405 = 3⁴ × 5¹
- 162 = 2¹ × 3⁴
Step 2: Identify Highest Powers
Now, let's identify the highest power of each prime factor present in the factorizations:
- The highest power of 2 is 2¹
- The highest power of 3 is 3⁴
- The highest power of 5 is 5¹
Step 3: Multiply Highest Powers
Finally, we multiply these highest powers together to find the LCM:
LCM(270, 405, 162) = 2¹ × 3⁴ × 5¹ = 2 × 81 × 5 = 810
Conclusion
The calculated LCM of 270, 405, and 162 is indeed 810. Therefore, statement a) is TRUE.
b) LCM(36, 64, 76) = 720
Next up, let's tackle statement b): LCM(36, 64, 76) = 720. We'll follow the same prime factorization method.
Step 1: Prime Factorization
Let's break down each number into its prime factors:
- 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2² × 3²
- 64 = 2 × 32 = 2 × 2 × 16 = 2 × 2 × 2 × 8 = 2 × 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶
- 76 = 2 × 38 = 2 × 2 × 19 = 2² × 19
So, we have:
- 36 = 2² × 3²
- 64 = 2⁶
- 76 = 2² × 19¹
Step 2: Identify Highest Powers
Now, we identify the highest power of each prime factor:
- The highest power of 2 is 2⁶
- The highest power of 3 is 3²
- The highest power of 19 is 19¹
Step 3: Multiply Highest Powers
Multiply these together to get the LCM:
LCM(36, 64, 76) = 2⁶ × 3² × 19¹ = 64 × 9 × 19 = 10944
Conclusion
The calculated LCM of 36, 64, and 76 is 10944, not 720. Therefore, statement b) is FALSE.
c) LCM(144, 96, 196) = 588
Alright, let's move on to statement c): LCM(144, 96, 196) = 588. Time for some more prime factorization!
Step 1: Prime Factorization
Let's find the prime factors of each number:
- 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3²
- 96 = 2 × 48 = 2 × 2 × 24 = 2 × 2 × 2 × 12 = 2 × 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 2 × 3 = 2⁵ × 3
- 196 = 2 × 98 = 2 × 2 × 49 = 2 × 2 × 7 × 7 = 2² × 7²
So, we have:
- 144 = 2⁴ × 3²
- 96 = 2⁵ × 3¹
- 196 = 2² × 7²
Step 2: Identify Highest Powers
Now, we identify the highest power of each prime factor:
- The highest power of 2 is 2⁵
- The highest power of 3 is 3²
- The highest power of 7 is 7²
Step 3: Multiply Highest Powers
Multiply these together to find the LCM:
LCM(144, 96, 196) = 2⁵ × 3² × 7² = 32 × 9 × 49 = 14112
Conclusion
The calculated LCM of 144, 96, and 196 is 14112, not 588. Therefore, statement c) is FALSE.
d) LCM(976, 490, 578) = 2928
Last but not least, let's look at statement d): LCM(976, 490, 578) = 2928. We're almost there, guys!
Step 1: Prime Factorization
Let's break each number down into its prime factors:
- 976 = 2 × 488 = 2 × 2 × 244 = 2 × 2 × 2 × 122 = 2 × 2 × 2 × 2 × 61 = 2⁴ × 61
- 490 = 2 × 245 = 2 × 5 × 49 = 2 × 5 × 7 × 7 = 2 × 5 × 7²
- 578 = 2 × 289 = 2 × 17 × 17 = 2 × 17²
So, we have:
- 976 = 2⁴ × 61¹
- 490 = 2¹ × 5¹ × 7²
- 578 = 2¹ × 17²
Step 2: Identify Highest Powers
Now, we identify the highest power of each prime factor:
- The highest power of 2 is 2⁴
- The highest power of 5 is 5¹
- The highest power of 7 is 7²
- The highest power of 17 is 17²
- The highest power of 61 is 61¹
Step 3: Multiply Highest Powers
Multiply these together to get the LCM:
LCM(976, 490, 578) = 2⁴ × 5¹ × 7² × 17² × 61¹ = 16 × 5 × 49 × 289 × 61 = 68336480
Conclusion
The calculated LCM of 976, 490, and 578 is a whopping 68,336,480, definitely not 2928. Therefore, statement d) is FALSE.
Final Results
Okay, guys, let's recap our findings:
- a) LCM(270, 405, 162) = 810 - TRUE
- b) LCM(36, 64, 76) = 720 - FALSE
- c) LCM(144, 96, 196) = 588 - FALSE
- d) LCM(976, 490, 578) = 2928 - FALSE
We did it! We successfully determined the truth value of each statement by calculating the LCM using the prime factorization method. Remember, math can be fun when we break it down step-by-step.
Key Takeaways:
- The LCM is the smallest number divisible by all given numbers.
- Prime factorization is a powerful tool for finding the LCM.
- Always double-check your calculations to avoid errors!
I hope this explanation was helpful, guys! Keep practicing, and you'll become LCM masters in no time!
