Problem 14 - Fundamentals of Mathematics

Question

Find the LCM of the following numbers. \(28,40,\) and 98

Step-by-Step Solution

Answer

The LCM of 28, 40, and 98 is 1960.

Step 1: Prime Factorization

First, find the prime factorization of each number. - For 28: \[ 28 = 2^2 \times 7 \]- For 40: \[ 40 = 2^3 \times 5 \]- For 98: \[ 98 = 2^1 \times 7^2 \]

Step 2: Identify Maximum Power of Each Prime

Identify the highest power of each prime number that appears in any of the factorizations:- The prime number 2 appears as the highest power in 40, i.e., \[ 2^3 \]- For the prime number 5, it appears as:\[ 5^1 \]- The prime number 7 appears as the highest power in 98, i.e., \[ 7^2 \]

Step 3: Calculate LCM

Multiply the highest powers of all prime numbers found in Step 2 to get the Least Common Multiple (LCM).\[ \text{LCM} = 2^3 \times 5^1 \times 7^2 \]

Step 4: Compute the LCM

Calculate the numerical value using the multiplication from Step 3:\[ \text{LCM} = 8 \times 5 \times 49 = 1960 \]

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 with no divisors other than 1 and themselves. When you do prime factorization, you express a number as a product of these prime numbers.
For example, consider the number 28. You can break it down as 28 = 2 x 14. Then, further factorize 14 as 2 x 7. Hence, 28 can be expressed as a product of prime numbers:

• 28 = 2² x 7

This process is useful for finding LCM because it shows all the prime components of the numbers involved.

Prime Numbers

Prime numbers play a crucial role in mathematics, especially in prime factorization. A prime number has exactly two distinct positive divisors: 1 and itself. They lay the foundation for constructing other numbers.
Here is a list of a few small prime numbers:

• 2
• 3
• 5
• 7
• 11

The prime number 2 is unique because it is the only even prime number. By using these fundamental pieces, you can easily express larger numbers.

Multiplication

Multiplication is the arithmetic operation of combining groups of equal size. For example, when calculating the LCM, multiplication helps combine the highest powers of primes found in the prime factorization process.
In the case of our numbers 28, 40, and 98, after identifying the highest powers of their prime factors, you multiply them together:

• \( 2^3 = 8 \)
• \( 5^1 = 5 \)
• \( 7^2 = 49 \)

Then, to find the LCM, you multiply these results: \( 8 \times 5 \times 49 = 1960 \). The multiplication step gives you the Least Common Multiple of the numbers.

Finding LCM

Finding the Least Common Multiple (LCM) involves identifying the smallest number divisible by each of the numbers in your set.
To find the LCM using prime factorization: 1. **Factorize each number into its prime factors.**2. **Choose the highest power of each prime number present across all factorizations.**3. **Multiply these highest powers together.** For example, for the numbers 28, 40, and 98, their LCM is calculated from:

• Highest power of 2: \( 2^3 \)
• Highest power of 5: \( 5^1 \)
• Highest power of 7: \( 7^2 \)

Thus, the LCM is \( 2^3 \times 5^1 \times 7^2 = 1960 \). This method ensures that the LCM is the smallest number that all original numbers can divide into without leaving a remainder.