Problem 28 - Fundamentals of Mathematics

Question

Find the least common multiple. 432 and 180

Step-by-Step Solution

Answer

The least common multiple of 432 and 180 is 2160.

Step 1: Prime Factorization of 432

Start by finding the prime factorization of 432. Divide repeatedly by the smallest prime numbers. 432 is even, so divide by 2: \[432 \div 2 = 216 \]Repeat: \[216 \div 2 = 108 \]\[108 \div 2 = 54 \]\[54 \div 2 = 27 \]Now, divide by 3, the smallest prime: \[27 \div 3 = 9 \]\[9 \div 3 = 3 \]\[3 \div 3 = 1 \]So, the prime factorization of 432 is: \[2^4 \times 3^3 \]

Step 2: Prime Factorization of 180

Next, find the prime factorization of 180 by dividing by the smallest prime numbers.180 is even, so start with 2: \[180 \div 2 = 90 \]\[90 \div 2 = 45 \]Now switch to the next smallest prime number, 3: \[45 \div 3 = 15 \]\[15 \div 3 = 5 \]Finally, 5 is a prime number: \[5 \div 5 = 1 \]Thus, the prime factorization of 180 is: \[2^2 \times 3^2 \times 5^1 \]

Step 3: Determine the LCM using the Highest Powers

To find the LCM, use the prime factors identified and take the highest power of each prime that appears in either factorization.- The highest power of 2 is \(2^4\).- The highest power of 3 is \(3^3\).- The highest power of 5 is \(5^1\).Thus, the LCM is calculated as:\[LCM = 2^4 \times 3^3 \times 5^1 \]

Step 4: Calculate the LCM

Now, perform the calculation to find the LCM.First, calculate \(2^4 \): \[2^4 = 16\]Next, calculate \(3^3 \):\[3^3 = 27\]Then calculate \(16 \times 27\):\[16 \times 27 = 432\]Finally, multiply by 5:\[432 \times 5 = 2160\]Hence, the LCM of 432 and 180 is 2160.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a whole number into its basic building blocks known as prime numbers. A prime number is a number greater than 1 with no divisors other than 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.

To perform prime factorization, you repeatedly divide the number by the smallest prime number possible until you're left with 1. For instance, with the number 432:

• Since 432 is even, start by dividing by 2: \(432 \div 2 = 216\).
• Continue dividing by 2: \(216 \div 2 = 108\), \(108 \div 2 = 54\), \(54 \div 2 = 27\).
• Now switch to the next smallest prime, 3: \(27 \div 3 = 9\), \(9 \div 3 = 3\), \(3 \div 3 = 1\).

This produces a prime factorization of: \(2^4 \times 3^3\).

Similarly, for 180, divide by the smallest prime numbers:

• Start with 2: \(180 \div 2 = 90\), \(90 \div 2 = 45\).
• Switch to 3: \(45 \div 3 = 15\), \(15 \div 3 = 5\), and finally \(5 \div 5 = 1\).

This results in the prime factorization of: \(2^2 \times 3^2 \times 5^1\).

Highest Power

When calculating the least common multiple (LCM), the concept of 'highest power' is crucial. It ensures that the LCM includes all necessary factors from the numbers involved. To obtain the highest power, look at each prime number identified in the factorization of both numbers.

For example, when determining the LCM of 432 and 180, check each prime factor's highest power:

• The highest power of 2 between \(2^4\) from 432 and \(2^2\) from 180 is \(2^4\).
• The highest power of 3 between \(3^3\) from 432 and \(3^2\) from 180 is \(3^3\).
• For prime 5, it occurs as \(5^1\) in 180. Since it doesn’t appear at all in 432, its highest power stays \(5^1\).

Selecting the highest power for each of these primes ensures the LCM fully encompasses all factors. This means when you multiply these powers together, you will cover both numbers efficiently without redundancy.

LCM Calculation

Calculating the least common multiple (LCM) involves multiplying the highest powers of all the prime factors that appear in the factorization of the numbers involved.

For our numbers 432 and 180, the LCM is derived from the identified highest powers:

• Take the highest power of 2, which is \(2^4 = 16\).
• Next, the highest power of 3 is \(3^3 = 27\).
• The highest power of 5 is \(5^1 = 5\).

Now, multiply these values together:

• First, calculate \(16 \times 27 = 432\).
• Then multiply 432 by 5: \(432 \times 5 = 2160\).

Thus, the LCM of 432 and 180 is 2160. This number is the smallest quantity that can be evenly divided by both 432 and 180, making it incredibly useful for solving problems involving fractions, synchronization, or finding common intervals.