HCF Of 45, 95, And 105: Easy Calculation Guide

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Hey guys! Today, we're going to break down how to find the HCF (Highest Common Factor) of 45, 95, and 105. This is a common math problem, and understanding how to solve it can really help you out. Let's dive in!

Understanding HCF

Before we jump into solving the problem, let's make sure we all know what HCF actually means. The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides evenly into each of the given numbers. Basically, it’s the biggest factor that all the numbers share. When dealing with numbers like 45, 95, and 105, finding the HCF helps simplify fractions, solve algebraic problems, and much more.

Why is HCF Important?

Understanding HCF isn't just about acing your math test. It has practical applications in everyday life. For instance, if you're trying to divide 45, 95, and 105 items into equal groups, the HCF will tell you the largest size each group can be. This is super useful in scenarios like event planning, resource allocation, and even in cooking when you need to scale recipes! Knowing how to find the HCF efficiently can save you time and effort in various tasks, making it a valuable skill to have. So, let's get started and see how we can easily find the HCF of 45, 95, and 105.

Methods to Find HCF

There are a couple of ways to find the HCF. We’ll explore two common methods: the prime factorization method and the division method. Each has its own advantages, and understanding both will give you a solid foundation for tackling HCF problems. The prime factorization method involves breaking down each number into its prime factors and then identifying the common ones. The division method, on the other hand, uses successive division to find the HCF. We’ll walk through each method step-by-step to make sure you get the hang of it.

Method 1: Prime Factorization

Prime factorization is a straightforward method to find the HCF. Here’s how you do it:

Step 1: Find the Prime Factors of Each Number

First, we need to break down each number (45, 95, and 105) into its prime factors. Prime factors are prime numbers that, when multiplied together, give you the original number. A prime number is a number that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).

  • Prime factors of 45: 3 x 3 x 5 = 3² x 5
  • Prime factors of 95: 5 x 19
  • Prime factors of 105: 3 x 5 x 7

Step 2: Identify Common Prime Factors

Now, let’s look at the prime factors we found and identify the ones that are common to all three numbers. In this case, we're looking for prime factors that appear in the factorization of 45, 95, and 105.

Looking at our factorizations:

  • 45 = 3² x 5
  • 95 = 5 x 19
  • 105 = 3 x 5 x 7

The only prime factor that all three numbers share is 5.

Step 3: Multiply the Common Prime Factors

Since we only have one common prime factor (which is 5), the HCF is simply that number. There's no need to multiply anything here because there's just one common factor.

Therefore, the HCF of 45, 95, and 105 is 5.

Quick Recap:

  1. Find prime factors: 45 = 3² x 5, 95 = 5 x 19, 105 = 3 x 5 x 7
  2. Identify common prime factors: 5
  3. Multiply common prime factors: 5

So, using the prime factorization method, we found that the HCF of 45, 95, and 105 is 5.

Method 2: Division Method

The division method, also known as Euclid's algorithm, is another effective way to find the HCF. This method involves successive division until you reach a remainder of 0. The last non-zero remainder is the HCF.

Step 1: Divide the Largest Number by the Smallest

First, identify the largest and smallest numbers from the set. In our case, the numbers are 45, 95, and 105. The largest number is 105, and the smallest is 45. Divide the largest number (105) by the smallest number (45).

105 ÷ 45 = 2 remainder 15

Step 2: Divide the Divisor by the Remainder

Now, take the divisor from the previous step (which was 45) and divide it by the remainder (which is 15).

45 ÷ 15 = 3 remainder 0

Since we've reached a remainder of 0, we stop here. The last non-zero remainder was 15.

Step 3: If More Than Two Numbers, Repeat the Process

We started with three numbers (45, 95, and 105), so we need to involve the remaining number (95) with the HCF we just found (15). Divide 95 by 15.

95 ÷ 15 = 6 remainder 5

Now, divide the divisor (15) by the remainder (5).

15 ÷ 5 = 3 remainder 0

Again, we've reached a remainder of 0. The last non-zero remainder was 5.

Therefore, the HCF of 45, 95, and 105 is 5.

Quick Recap:

  1. Divide 105 by 45: 105 ÷ 45 = 2 remainder 15
  2. Divide 45 by 15: 45 ÷ 15 = 3 remainder 0 (HCF of 105 and 45 is 15)
  3. Divide 95 by 15: 95 ÷ 15 = 6 remainder 5
  4. Divide 15 by 5: 15 ÷ 5 = 3 remainder 0 (HCF of 95 and 15 is 5)

So, using the division method, we also found that the HCF of 45, 95, and 105 is 5.

Comparing the Methods

Both the prime factorization method and the division method are effective for finding the HCF, but they have their own pros and cons.

  • Prime Factorization: This method is great for understanding the underlying factors of the numbers. It’s straightforward and easy to visualize, especially for smaller numbers. However, it can become cumbersome with larger numbers that have many prime factors. Prime factorization is also excellent for finding both the HCF and the LCM (Least Common Multiple) simultaneously.
  • Division Method: This method is particularly useful for larger numbers where finding prime factors might be challenging. It’s a systematic approach that reduces the numbers step-by-step until you find the HCF. The division method is efficient and doesn’t require you to know the prime factors, making it a good choice when you’re working with complex numbers.

In the case of 45, 95, and 105, both methods work well, but the division method might be slightly quicker since you don’t need to find all the prime factors.

Conclusion

Alright, guys! We’ve walked through two different methods to find the HCF of 45, 95, and 105. Whether you prefer breaking down numbers into their prime factors or using successive division, the key takeaway is that the HCF of these three numbers is 5.

Understanding HCF is super useful in many areas, from simplifying math problems to real-life applications like dividing items into equal groups. So, keep practicing, and you’ll become a pro at finding the HCF in no time! Keep up the great work, and happy calculating!