NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 3(ii)

NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 3(ii)

Question:

Find the LCM and HCF of the following integers by applying the prime factorisation method: 17, 23, and 29.

Given:

The integers are 17, 23, and 29.

To Find:

Find the LCM and HCF of 17, 23, and 29 using the prime factorisation method.

Formula:

  • For any set of numbers:
    • HCF is the product of the smallest powers of all common prime factors.
    • LCM is the product of the highest powers of all prime factors present in any of the numbers.

For distinct prime numbers \(p, q, r\):

  • $$\text{HCF}(p, q, r) = 1$$
  • $$\text{LCM}(p, q, r) = p \times q \times r$$

Solution:

Step 1: Write the prime factorisation of each number.

  • 17 is a prime number: $$17 = 17^1$$
  • 23 is a prime number: $$23 = 23^1$$
  • 29 is a prime number: $$29 = 29^1$$

Step 2: Find the HCF (Highest Common Factor).

There are no common prime factors among 17, 23, and 29.

Therefore,

$$\text{HCF}(17,\,23,\,29) = 1$$

Step 3: Find the LCM (Lowest Common Multiple).

Take the product of all the unique prime factors:

$$\text{LCM}(17,\,23,\,29) = 17 \times 23 \times 29$$

Step 4: Calculate the value of the LCM.

First, multiply 17 and 23:

$$17 \times 23 = 391$$

Now, multiply the result by 29:

$$391 \times 29 = 11339$$

So,

$$\text{LCM}(17,\,23,\,29) = 11339$$

Result:

  • $$\text{HCF}(17,\,23,\,29) = 1$$
  • $$\text{LCM}(17,\,23,\,29) = 11339$$
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