NCERT Class X Chapter 1: Real Numbers Example 4

NCERT Class X Chapter 1: Real Numbers Example 4

Question:

Find the HCF and LCM of 6, 72, and 120 using the prime factorisation method.

Given:

The numbers are 6, 72, and 120.

To Find:

The Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 6, 72, and 120 using the prime factorisation method.

Formula:

  • HCF (by prime factorisation): Multiply the lowest powers of all common prime factors present in each number.
  • LCM (by prime factorisation): Multiply the highest powers of all prime factors present in any of the numbers.

Solution:

Step 1: Write the prime factorisation of each number.

$$6 = 2 \times 3$$ $$72 = 2 \times 2 \times 2 \times 3 \times 3$$ $$120 = 2 \times 2 \times 2 \times 3 \times 5$$

Step 2: Express each number in exponential form.

$$6 = 2^1 \times 3^1$$ $$72 = 2^3 \times 3^2$$ $$120 = 2^3 \times 3^1 \times 5^1$$

Step 3: Find the HCF by taking the lowest power of all common prime factors.

  • For 2: Exponents are 1 (in 6), 3 (in 72), 3 (in 120). Lowest is 1.
  • For 3: Exponents are 1 (in 6), 2 (in 72), 1 (in 120). Lowest is 1.
  • For 5: Not present in all numbers, so not included.
$$\text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6$$

Step 4: Find the LCM by taking the highest power of all prime factors present in any number.

  • For 2: Highest power is 3.
  • For 3: Highest power is 2.
  • For 5: Highest power is 1.
$$\text{LCM} = 2^3 \times 3^2 \times 5^1$$

Step 5: Calculate the value of the LCM.

$$2^3 = 8$$ $$3^2 = 9$$ $$8 \times 9 = 72$$ $$72 \times 5 = 360$$ $$\text{LCM} = 360$$

Result:

  • HCF(6, 72, 120) = 6
  • LCM(6, 72, 120) = 360
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