Find the HCF and LCM of 336 and 54 by the prime factorization method, and verify that $ ext{HCF} imes ext{LCM} =

Question

Find the HCF and LCM of 336 and 54 by the prime factorization method, and verify that $	ext{HCF} 	imes 	ext{LCM} = 	ext{Product of the two numbers}$.

Accepted Answer

To find the HCF and LCM of 336 and 54 using the prime factorization method, we first find the prime factors of each number. 
Step 1: Find the prime factorization of 336. $$336 = 2 	imes 168 = 2 	imes 2 	imes 84 = 2 	imes 2 	imes 2 	imes 42 = 2 	imes 2 	imes 2 	imes 2 	imes 21 = 2^4 	imes 3^1 	imes 7^1$$ 
Step 2: Find the prime factorization of 54. $$54 = 2 	imes 27 = 2 	imes 3 	imes 3 	imes 3 = 2^1 	imes 3^3$$ 
Step 3: Calculate the HCF.
The HCF is the product of the smallest power of each common prime factor in the numbers. The common prime factors are 2 and 3. $$	ext{HCF}(336, 54) = 2^1 	imes 3^1 = 2 	imes 3 = 6$$ 
Step 4: Calculate the LCM.
The LCM is the product of the greatest power of each prime factor involved in the numbers. $$	ext{LCM}(336, 54) = 2^4 	imes 3^3 	imes 7^1 = 16 	imes 27 	imes 7 = 432 	imes 7 = 3024$$ 
Step 5: Verify the relation $	ext{HCF} 	imes 	ext{LCM} = 	ext{Product of the two numbers}$. $$	ext{HCF} 	imes 	ext{LCM} = 6 	imes 3024 = 18144$$ $$	ext{Product of the two numbers} = 336 	imes 54 = 18144$$ 
Since $	ext{HCF} 	imes 	ext{LCM} = 	ext{Product of the two numbers}$ ($18144 = 18144$), the given relation is verified.

Real Numbers

❓ Sibling Questions in this Chapter