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Chapter 3: Problem 5

Find the LCM of each set of numbers. $$ 20,40 $$

Short Answer

Expert verified
The LCM of 20 and 40 is 40.

Step by step solution

01

- Prime Factorization

Find the prime factorization of each number.\[ 20 = 2^2 \times 5 \] \[ 40 = 2^3 \times 5 \]
02

- Identify Highest Powers

Identify the highest powers of all prime numbers in the factorizations. The prime numbers are 2 and 5. The highest powers are: \[ 2^3 \text{ (from 40)} \] \[ 5^1 \text{ (from 20 and 40)} \]
03

- Calculate the LCM

Multiply the highest powers to find the LCM.\[ LCM = 2^3 \times 5^1 = 8 \times 5 = 40 \]

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into its basic building blocks鈥攑rime numbers.
A prime number is a number that can only be divided by 1 and itself.
For instance, consider the numbers 20 and 40.
Let鈥檚 write down their prime factors:
- For 20, the prime factors are \(2^2 \times 5\).
- For 40, the prime factors are \(2^3 \times 5\).
Notice that both numbers share the prime factors 2 and 5.
These prime factors will be used in the next step to find the least common multiple (LCM).
Highest Powers
After prime factorization, we need to identify the highest powers of all prime numbers found in the factorizations.
This step ensures that the LCM calculation includes all the necessary factors.
- From 20, we have the prime factors \(2^2 \) and \(5^1 \).
- From 40, we have the prime factors \(2^3\) and \( 5^1\).
The highest power of each prime is:
- For the prime number 2, the highest power is \(2^3 \) (from 40).
- For the prime number 5, the highest power is \( 5^1\) (from both 20 and 40).
These highest powers are crucial for the LCM calculation.
LCM Calculation
The final step is to calculate the least common multiple (LCM) using the highest powers identified.
To do this, simply multiply the highest powers of all prime numbers involved.
- Here, the highest powers are \( 2^3 \) and \( 5^1 \) from the previous step.
The LCM is calculated as follows:
\[ LCM = 2^3 \times 5^1 = 8 \times 5 = 40 \]
Therefore, the LCM of 20 and 40 is 40.
This method ensures you capture all prime factors at their highest powers, resulting in the smallest number that both original numbers can divide into evenly.

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