Question

HCF of two co primes (say \(x\) and \(y\) ) is (1) \(\mathrm{x}\) (2) \(\mathrm{y}\) (3) \(\mathrm{xy}\) (4) 1

Short answer

Answer: 1

Step-by-step solution

Recall the definition of co-prime numbers

Co-prime numbers are two numbers that have no common factor other than 1. In other words, the greatest factor that they share is 1.

Apply the definition to the given numbers x and y

We are told that x and y are co-prime. By definition, their highest common factor must be 1.

Choose the correct option

We are given four options: (1) x (2) y (3) xy (4) 1 Since the HCF of two co-prime numbers is 1, the correct answer is option (4) 1.

Key concepts

Co-Prime Numbers

Co-prime numbers, also known as relatively prime or mutually prime numbers, are pairs of numbers that hold a special relationship in mathematics. They only share one divisor: the number 1. This means that if you were to list out all the factors of each number in the pair, the only common factor you would find is 1. For example, consider the numbers 8 and 15. The factors of 8 are 1, 2, 4, and 8, while the factors of 15 are 1, 3, 5, and 15. As you can see, they have no other common factors besides 1, making them co-prime.

Understanding this concept is crucial because it forms the basis for many theorems and algorithms in number theory, such as those concerning the simplification of fractions or the operation of certain encryption algorithms. It's also remarkably simple to grasp that two numbers don't need to be prime themselves to be co-prime; they only need to lack any common factors other than 1.

Highest Common Factor

The highest common factor (HCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more integers without leaving a remainder. For understanding HCF, imagine it as the 'biggest piece' of a puzzle that perfectly fits into the 'spaces' of two numbers. In our exercise, the confusion might arise when we think of the product, sum, or individual numbers as potential candidates for the HCF when, in fact, the concept of co-primality overrides these possibilities. For co-prime numbers, such as the pair mentioned in our problem, their HCF will always be 1, because they do not share any larger factors.

Calculating the HCF can be done through various methods, such as listing out the factors, employing the Euclidean algorithm, or by using prime factorization. However, in the case of co-prime numbers, the mathematics simplifies beautifully, creating an elegant property that allows us to instantly know the HCF without any need for calculations.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that enables students to apply mathematical concepts to real-world scenarios. In this type of problem-solving, understanding the definitions and relationships between mathematical concepts leads to the solution. As demonstrated in the provided exercise, the process began by recalling the definition of co-prime numbers. Once that was established, the relationship between the given numbers, in this case, x and y, was analyzed based on that definition.

If you're facing a mathematical problem, a structured approach can be your best ally:

• Identify the core mathematical concepts relevant to the problem.
• Apply these concepts to the specifics of the problem.
• Consider the logic and properties that dictate the relationships between the numbers or elements involved.
• Carefully analyze each potential answer with respect to the definitions and properties you have recalled.

Such a methodology not only leads to the correct solution, as it did with selecting option 4 (1) for the HCF of co-prime numbers, but it also reinforces the student's understanding of core mathematical concepts, equipping them with tools necessary for future and more complex mathematical challenges.