Problem 69 - Fundamentals of Mathematics

Question

(Section 3.6) Find the least common multiple of 28 and 36 .

Step-by-Step Solution

Answer

The LCM of 28 and 36 is 252.

Step 1: Prime Factorization of 28

To find the least common multiple (LCM) of two numbers, we first need their prime factorizations. Start with the number 28. The prime factorization of 28 is determined as follows:- 28 is divisible by 2 (smallest prime number): \[ 28 \div 2 = 14 \]- 14 is also divisible by 2: \[ 14 \div 2 = 7 \]- 7 is a prime number.Thus, the prime factorization of 28 is: \[ 28 = 2^2 \times 7 \]

Step 2: Prime Factorization of 36

Next, find the prime factorization of 36:- 36 is divisible by 2: \[ 36 \div 2 = 18 \]- 18 is divisible by 2: \[ 18 \div 2 = 9 \]- 9 is divisible by 3: \[ 9 \div 3 = 3 \]- 3 is a prime number.Thus, the prime factorization of 36 is:\[ 36 = 2^2 \times 3^2 \]

Step 3: LCM Calculation Using Prime Factors

To find the LCM, use the highest power of each prime number from the factorizations:- For the prime number 2: the maximum power is 2 (from both 28 and 36).- For the prime number 3: the maximum power is 2 (from 36).- For the prime number 7: the maximum power is 1 (from 28).Multiply these together to get the LCM:\[ \text{LCM} = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 \]

Step 4: Simplify the Product

Now, simplify the product to find the final LCM:Calculate step by step:- \( 4 \times 9 = 36 \)- \( 36 \times 7 = 252 \)Therefore, the LCM of 28 and 36 is 252.

Key Concepts

Prime Factorization

Prime factorization is an important concept in mathematics that involves breaking down a number into a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, etc. To perform a prime factorization, repeatedly divide the number by the smallest possible prime numbers until only prime numbers are left as factors.

For the example of finding the least common multiple (LCM) of 28 and 36, we start by breaking each number down into its prime factors:

• 28 is divided by 2 to give 14, divided by 2 again to give 7. Since 7 is already a prime number, the prime factorization of 28 is: \[ 28 = 2^2 \times 7 \]
• 36 is divided by 2 to give 18, divided by 2 again to give 9, and finally divided by 3 to give 3. The prime factorization of 36 is: \[ 36 = 2^2 \times 3^2 \]

This method is useful for uncovering the building blocks of numbers and is a key step in determining features like the LCM.

Greatest Power

In the context of least common multiple calculations, understanding the greatest power of each prime factor is crucial. Once you have the prime factorization of multiple numbers, you must identify the highest power of each prime number appearing in the factorizations.

Let's see how this applies:

• From the factorization of 28: \(2^2\) and \(7^1\)
• From the factorization of 36: \(2^2\) and \(3^2\)

To determine the LCM, consider each prime number across all numbers' factorizations:

• For the prime number 2, the greatest power is \(2^2\).
• For 3, the greatest power is \(3^2\).
• And for 7, the greatest power is \(7^1\).

By selecting the highest power, you ensure that any multiple of the LCM is also a multiple of both numbers.

Multiplicative Properties

The multiplicative properties in mathematics refer to how numbers are combined through multiplication. When calculating the least common multiple using prime factorization, you multiply the greatest powers of each found prime together.

This enables you to compile all essential factors of the original numbers. Consider the product:

• For our current LCM calculation: \[ \text{LCM} = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 \]
• Simplifying that gives the final product as: \\[ 4 \times 9 = 36 \]
• Followed by \( 36 \times 7 = 252 \)

This result, 252, is the LCM of 28 and 36.The beauty of the multiplicative properties lies in how you can confidently calculate the smallest common denominator for any two integers, leading to manifold applications from fraction arithmetic to solving equations.