HCF Of 420 & 1176 And Square Number Division Of 1176

Hey guys! Let's dive into an interesting math problem involving prime factorization, the highest common factor (HCF), and finding the smallest divisor to create a perfect square. We'll break down the problem step by step, making sure everyone can follow along. So, buckle up and let's get started!

Understanding Prime Factorization and HCF

Our journey begins with understanding prime factorization and how it helps us find the HCF. Prime factorization is like breaking down a number into its fundamental building blocks – prime numbers. Remember, a prime number is a number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, and so on). When we express a number as a product of its prime factors, we gain a deeper understanding of its composition, which is super useful for solving various mathematical problems, including finding the HCF.

In this particular problem, we are given that 1176 can be written as $2^3 imes 3 imes 7^2$. This means 1176 is made up of three 2s, one 3, and two 7s multiplied together. This prime factorization is our starting point. Now, to find the HCF of two numbers, we need to identify the common prime factors and their lowest powers present in both numbers. The HCF, also known as the greatest common divisor (GCD), is the largest number that divides both numbers without leaving a remainder. It's a fundamental concept in number theory and has practical applications in various fields, such as simplifying fractions and solving real-world problems involving sharing or grouping items.

So, how do we find the HCF using prime factorization? First, we express both numbers in their prime factorized forms. Then, we identify the prime factors that are common to both numbers. For each common prime factor, we take the lowest power that appears in either factorization. Finally, we multiply these common prime factors raised to their lowest powers to get the HCF. This method provides a systematic and efficient way to determine the HCF, especially for larger numbers where traditional methods might be cumbersome. Understanding prime factorization and HCF is not just about solving this specific problem; it's about building a solid foundation in number theory, which is essential for tackling more advanced mathematical concepts.

Part (a): Finding the Highest Common Factor of 420 and 1176

Let's tackle the first part of our problem: finding the highest common factor (HCF) of 420 and 1176. We already know the prime factorization of 1176, which is $2^3 imes 3 imes 7^2$. Now, we need to find the prime factorization of 420. To do this, we can use a factor tree or division method. Let's use the division method:

• 420 ÷ 2 = 210
• 210 ÷ 2 = 105
• 105 ÷ 3 = 35
• 35 ÷ 5 = 7
• 7 ÷ 7 = 1

So, the prime factorization of 420 is $2^2 imes 3 imes 5 imes 7$. Now we have the prime factorizations of both numbers:

• 1176 = $2^3 imes 3 imes 7^2$
• 420 = $2^2 imes 3 imes 5 imes 7$

To find the HCF, we identify the common prime factors and their lowest powers. The common prime factors are 2, 3, and 7. The lowest powers are:

• 2: The lowest power is $2^2$
• 3: The lowest power is $3^1$
• 7: The lowest power is $7^1$

Now, we multiply these together: HCF = $2^2 imes 3 imes 7 = 4 imes 3 imes 7 = 84$. Therefore, the highest common factor of 420 and 1176 is 84. This means that 84 is the largest number that divides both 420 and 1176 without leaving a remainder. Finding the HCF is a crucial skill in various mathematical contexts, such as simplifying fractions and solving problems involving ratios and proportions.

Part (b): Finding the Smallest Divisor for a Square Number

Now, let's move on to the second part of the problem: finding the smallest whole number by which 1176 can be divided to give a square number. Remember, a square number is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). To solve this, we again rely on the prime factorization of 1176, which is $2^3 imes 3 imes 7^2$.

To get a square number, all the exponents in the prime factorization must be even. Currently, we have:

• 2: Exponent is 3 (odd)
• 3: Exponent is 1 (odd)
• 7: Exponent is 2 (even)

We need to make the exponents of 2 and 3 even. To do this, we can divide 1176 by the factors that have odd exponents. In this case, we need to divide by $2^1$ and $3^1$. So, the smallest whole number we need to divide by is $2 imes 3 = 6$. Let's verify this. If we divide 1176 by 6, we get:

$$1176 ÷ 6 = 196$$

Now, let's find the prime factorization of 196:

• 196 ÷ 2 = 98
• 98 ÷ 2 = 49
• 49 ÷ 7 = 7
• 7 ÷ 7 = 1

So, the prime factorization of 196 is $2^2 imes 7^2$. Notice that both exponents are even, which means 196 is indeed a square number (196 = 14^2). Therefore, the smallest whole number by which 1176 can be divided to give a square number is 6. This concept is closely related to finding the square root of a number and understanding the properties of perfect squares. It's a valuable skill in algebra and other areas of mathematics.

Wrapping Up: Key Takeaways

Alright, guys, we've successfully tackled this problem! We found the highest common factor of 420 and 1176 to be 84, and we determined that 1176 can be divided by 6 to yield a square number. The key to solving this problem was understanding prime factorization and how it relates to HCF and square numbers. Remember, prime factorization is a powerful tool that can simplify many mathematical problems.

By breaking down numbers into their prime factors, we can easily identify common factors, determine HCFs, and find divisors that result in square numbers. These are fundamental concepts in number theory and have applications in various areas of mathematics and beyond. Keep practicing these concepts, and you'll become a math whiz in no time! Remember, math is not just about memorizing formulas; it's about understanding the underlying principles and applying them creatively to solve problems.