What will be obtained if 8 is Subtracted from the HCF of 168, 189, and 231?

What will be obtained if 8 is Subtracted from the HCF of 168, 189, and 231?

If you want to know what will be obtained if 8 is subtracted from the HCF of 168, 189, and 231, follow these steps:

Prime factorization of 168 → 2³ x 3 x 7

Prime factorization of 189 → 3³ x 7

Prime factorization of 231 → 3 x 7 x 11

The common factors among all three numbers are 3 x 7

Therefore, the HCF of 168, 189, and 231 is 3 x 7 = 21

Now, subtract 8 from the HCF obtained

21 − 8 = 13

Therefore, if 8 is subtracted from the HCF of 168, 189, and 231, the result is 13.

Highest Common Factor (HCF) and its Applications

The HCF or GCD of two or more numbers is the largest positive integer that divides each of them without leaving a remainder. It's a fundamental concept in arithmetic and number theory.

Finding Common Divisors:

To determine the HCF/GCD of numbers, mathematicians look for the highest number that evenly divides all given numbers. This involves identifying common factors shared by all numbers and finding their greatest value.

Prime Factorization Method:

One method to find the HCF/GCD involves prime factorization. By breaking down each number into its prime factors and identifying common factors, we can determine the product of the smallest powers of all primes present in all numbers.

Relationship with LCM:

The HCF/GCD and Least Common Multiple (LCM) are related concepts. The product of the HCF and LCM of two numbers equals the product of the numbers themselves. This relationship is used to solve equations involving multiple variables.

Application

• HCF/GCD is used to simplify fractions. By dividing both the numerator and denominator of a fraction by their HCF/GCD, we can reduce the fraction to its simplest form.
• Beyond mathematics, understanding HCF/GCD has practical applications in fields such as computer science, engineering, and economics.
• Learning about HCF/GCD enhances mathematical reasoning and problem-solving skills. It teaches students how to analyze number relationships and apply divisibility rules effectively.
• In mathematics and beyond, HCF/GCD is crucial for solving problems involving distribution, allocation, and optimization.