Let A And B Be Two Consecutive Natural Numbers Such That A = P Q³ And B = P³q², Where P And Q Are Prime Numbers. If HCF(a, B) = Pmq And LCM (a, B) = Prqs Then (m + N)-(r+s) Is (a) 2 (b) -2 (c) 5 (d) 7

Mar 8, 2025 by ADMIN 203 views

Let a and b be two consecutive natural numbers such that a = p q³ and b = p³q², where p and q are prime numbers.

Introduction

In this discussion, we will explore the relationship between two consecutive natural numbers, a and b, where a is expressed as p q³ and b is expressed as p³q². Here, p and q are prime numbers. We will determine the highest common factor (HCF) and the least common multiple (LCM) of a and b, and then use these values to find the expression (m + n) - (r + s).

Understanding the HCF and LCM

The HCF of two numbers is the greatest number that divides both of them without leaving a remainder. On the other hand, the LCM of two numbers is the smallest number that is a multiple of both of them.

Finding the HCF of a and b

To find the HCF of a and b, we need to identify the common factors of a and b. Since a = p q³ and b = p³q², we can see that both a and b have the common factors p and q.

import Data.List
-- Define the prime numbers p and q
p = 2
q = 3
-- Define the numbers a and b
a = p * (q ^ 3)
b = (p ^ 3) * (q ^ 2)
-- Find the HCF of a and b
hcf = gcd a b

In this code, we use the gcd function from the Data.List module to find the greatest common divisor of a and b, which is the HCF.

Finding the LCM of a and b

To find the LCM of a and b, we need to identify the smallest number that is a multiple of both a and b. Since a = p q³ and b = p³q², we can see that the LCM of a and b is the product of the highest powers of p and q that divide both a and b.

-- Find the LCM of a and b
lcm = (a * b) `div` hcf

In this code, we use the formula for the LCM, which is the product of the two numbers divided by their HCF.

Determining the Values of m, n, r, and s

From the given information, we know that HCF(a, b) = pmq" and LCM(a, b) = prqs. We can see that the HCF is the product of the common factors p and q, and the LCM is the product of the highest powers of p and q that divide both a and b.

-- Define the values of m, n, r, and s
m = 1
n = 3
r = 3
s = 2

In this code, we define the values of m, n, r, and s based on the given information.

Finding the Expression (m + n) - (r + s)

Now that we have the values of m, n, r, and s, we can find the expression (m + n) - (r + s).

-- Find the expression (m + n) - (r + s)
result = (m + n) - (r + s)

In this code, we use the values of m, n, r, and s to find the expression (m + n) - (r + s).

Conclusion

In this discussion, we explored the relationship between two consecutive natural numbers, a and b, where a is expressed as p q³ and b is expressed as p³q². We determined the HCF and LCM of a and b, and then used these values to find the expression (m + n) - (r + s). The final answer is (m + n) - (r + s) = 2.

Final Answer

The final answer is (m + n) - (r + s) = 2.

Introduction

In our previous discussion, we explored the relationship between two consecutive natural numbers, a and b, where a is expressed as p q³ and b is expressed as p³q². We determined the HCF and LCM of a and b, and then used these values to find the expression (m + n) - (r + s). In this Q&A article, we will address some common questions related to this topic.

Q: What are the prime numbers p and q?

A: The prime numbers p and q are the building blocks of the numbers a and b. In this case, we can choose any two prime numbers, but for simplicity, let's assume p = 2 and q = 3.

Q: What is the HCF of a and b?

A: The HCF of a and b is the greatest number that divides both a and b without leaving a remainder. In this case, the HCF is pmq", where m and n are the powers of p and q that divide both a and b.

Q: What is the LCM of a and b?

A: The LCM of a and b is the smallest number that is a multiple of both a and b. In this case, the LCM is prqs, where r and s are the powers of p and q that divide both a and b.

Q: How do we find the values of m, n, r, and s?

A: To find the values of m, n, r, and s, we need to identify the common factors of a and b. Since a = p q³ and b = p³q², we can see that both a and b have the common factors p and q. The powers of p and q that divide both a and b are m = 1, n = 3, r = 3, and s = 2.

Q: What is the expression (m + n) - (r + s)?

A: The expression (m + n) - (r + s) is a mathematical expression that involves the values of m, n, r, and s. In this case, the expression is (m + n) - (r + s) = (1 + 3) - (3 + 2) = 2.

Q: What is the significance of the expression (m + n) - (r + s)?

A: The expression (m + n) - (r + s) is a mathematical expression that can be used to find the difference between the powers of p and q that divide both a and b. In this case, the expression is equal to 2, which means that the difference between the powers of p and q that divide both a and b is 2.

Q: Can we use this method to find the HCF and LCM of any two numbers?

A: Yes, we can use this method to find the HCF and LCM of any two numbers. However, we need to identify the common factors of the two numbers and find the powers of those factors that divide both numbers.

Q: What are some real-world applications of finding the HCF and LCM of two numbers?

A: Finding the HCF and LCM of two numbers has many real-world applications, such as:

Finding the greatest common divisor of two numbers to determine the maximum number of items that can be shared among a group of people.
Finding the least common multiple of two numbers to determine the smallest number that is a multiple of both numbers.
Finding the HCF and LCM of two numbers to determine the difference between the powers of prime factors that divide both numbers.

Conclusion

In this Q&A article, we addressed some common questions related to finding the HCF and LCM of two consecutive natural numbers, a and b, where a is expressed as p q³ and b is expressed as p³q². We also discussed some real-world applications of finding the HCF and LCM of two numbers.