Problem 12 - Basic Engineering Mathematics

Question

d etermine the HCF of the numbers 30,105 , 210 and 1155

Step-by-Step Solution

Answer

The HCF of 30, 105, 210, and 1155 is 15.

Step 1: Prime Factorization of 30

Begin by finding the prime factorization of the number 30. Divide 30 by the smallest prime number, which is 2, to get 15. Since 15 is not divisible by 2, proceed to the next smallest prime, which is 3. Divide 15 by 3 to get 5. Thus, the prime factorization of 30 is \(2 \times 3 \times 5\).

Step 2: Prime Factorization of 105

Next, find the prime factorization of 105. Start by dividing 105 by the smallest prime number, 3, to get 35. Since 35 is not divisible by 3, try the next smallest prime number, which is 5. Divide 35 by 5 to get 7. Therefore, the prime factorization of 105 is \(3 \times 5 \times 7\).

Step 3: Prime Factorization of 210

Proceed with the prime factorization of 210. Start by dividing 210 by the smallest prime, 2, to get 105. Next, divide 105 by 3 to get 35, then divide 35 by 5 to get 7. So, the prime factorization of 210 is \(2 \times 3 \times 5 \times 7\).

Step 4: Prime Factorization of 1155

Find the prime factorization of 1155. Start with 3 (as 1155 is odd), divide 1155 by 3 to get 385. Now, divide 385 by 5 to get 77. Then, divide 77 by 7 to get 11. Finally, 11 is a prime number. Thus, the prime factorization is \(3 \times 5 \times 7 \times 11\).

Step 5: Identify Common Prime Factors

Identify the common prime factors from all the numbers: 30 \((2 \times 3 \times 5)\), 105 \((3 \times 5 \times 7)\), 210 \((2 \times 3 \times 5 \times 7)\), and 1155 \((3 \times 5 \times 7 \times 11)\). The common prime factors are 3 and 5.

Step 6: Calculate the HCF

Multiply the common prime factors together to find the HCF. Only the numbers 3 and 5 appear in each factorization, so the HCF is \(3 \times 5 = 15\).

Key Concepts

Prime Factorization

Prime factorization is a method used in mathematics to express a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This method is vital when solving problems related to the Highest Common Factor (HCF), as it helps to break down complex numbers into simpler components.

To perform prime factorization, you can follow these steps:

• Start with the smallest prime number, which is 2.
• Divide the number by the prime number - if it divides evenly, continue dividing by that prime until it no longer does.
• Move on to the next smallest prime number and repeat the process.
• Continue this until the number has been completely factorized into a series of prime numbers.

In our example, the numbers 30, 105, 210, and 1155 were factorized into their prime components. Such an approach makes it easier to identify the factors that these numbers share in common.

Common Prime Factors

Common prime factors refer to prime numbers that appear in the prime factorizations of two or more numbers. Identifying these is crucial in solving for the HCF since only these shared primes will be used to calculate the HCF.

In our example, after performing prime factorizations:

• 30 becomes: \(2 \times 3 \times 5\)
• 105 becomes: \(3 \times 5 \times 7\)
• 210 becomes: \(2 \times 3 \times 5 \times 7\)
• 1155 becomes: \(3 \times 5 \times 7 \times 11\)

The numbers 3 and 5 are the common prime factors visible in all their prime factorizations. Finding these common factors is the key to determining the HCF, as they are the numbers the original numbers "agree" upon in their prime forms.

Mathematics Problem-Solving

Mathematics problem-solving often involves systematically breaking down numbers into their fundamental parts to address complex questions. Solving for the Highest Common Factor (HCF) using prime factorization is a good example of this.

This method:

• Helps to simplify large numbers into manageable building blocks (prime numbers).
• Uses logic and systematic approaches to handle potential errors in calculations.
• Allows for efficient identification of patterns through the breakdown of complex data.

By identifying the HCF of 30, 105, 210, and 1155, the exercise demonstrates how breaking numbers into their prime components and identifying commonalities can simplify and solve what might initially seem to be complex problems.

Problem-solving in mathematics hence becomes an exercise in transforming complexity into simplicity, ensuring clarity and understanding in mathematical processes.