CBSE · Class 10 · Mathematics · Real NumbersFind the LCM and HCF of 336 and 54 by the prime factorization method, and verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$.

Step 1: Prime Factorization of the Given Numbers\nTo find the LCM and HCF of 336 and 54, we first express each number as a product of its prime factors using factor trees or division.

Prime factorization of 336:

• $336 \div 2 = 168$
• $168 \div 2 = 84$
• $84 \div 2 = 42$
• $42 \div 2 = 21$
• $21 \div 3 = 7$
• $7 \div 7 = 1$ \nThus, the prime factorization of $336$ is: $$336 = 2 \times 2 \times 2 \times 2 \times 3 \times 7 = 2^4 \times 3^1 \times 7^1$$

Prime factorization of 54:

• $54 \div 2 = 27$
• $27 \div 3 = 9$
• $9 \div 3 = 3$
• $3 \div 3 = 1$ \nThus, the prime factorization of $54$ is: $$54 = 2 \times 3 \times 3 \times 3 = 2^1 \times 3^3$$

Step 2: Calculation of HCF\nThe Highest Common Factor (HCF) is the product of the smallest power of each common prime factor involved in the numbers.

• Common prime factors of 336 and 54 are $2$ and $3$.
• The smallest power of $2$ is $2^1$.
• The smallest power of $3$ is $3^1$.

$$\text{HCF}(336, 54) = 2^1 \times 3^1 = 2 \times 3 = 6$$

Step 3: Calculation of LCM\nThe Least Common Multiple (LCM) is the product of the greatest power of each prime factor involved in the numbers.

• Prime factors involved are $2$, $3$, and $7$.
• The greatest power of $2$ is $2^4$.
• The greatest power of $3$ is $3^3$.
• The greatest power of $7$ is $7^1$.

$$\text{LCM}(336, 54) = 2^4 \times 3^3 \times 7^1$$ $$\text{LCM}(336, 54) = 16 \times 27 \times 7$$ $$16 \times 27 = 432$$ $$432 \times 7 = 3024$$ $$\text{LCM}(336, 54) = 3024$$

Step 4: Verification of the Formula\nWe need to verify the property: $\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$, where $a = 336$ and $b = 54$.

Calculate Left Hand Side (LHS): $$\text{LHS} = \text{LCM} \times \text{HCF} = 3024 \times 6 = 18144$$

Calculate Right Hand Side (RHS): $$\text{RHS} = \text{Product of the two numbers} = 336 \times 54$$ $$336 \times 50 = 16800$$ $$336 \times 4 = 1344$$ $$16800 + 1344 = 18144$$ $$\text{RHS} = 18144$$

Final Conclusion:\nSince $\text{LHS} = \text{RHS} = 18144$, the relation $\text{LCM} \times \text{HCF} = \text{Product of two numbers}$ is successfully verified.