Find the HCF of 96 and 404 by prime factorisation method. Hence find their LCM.
The Correct Answer and Explanation is:
To find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of 96 and 404 using the prime factorization method, follow these steps:
Step 1: Prime Factorization of 96
Start by dividing 96 by the smallest prime number (2) until you can’t divide anymore.96÷2=4896 \div 2 = 4896÷2=4848÷2=2448 \div 2 = 2448÷2=2424÷2=1224 \div 2 = 1224÷2=1212÷2=612 \div 2 = 612÷2=66÷2=36 \div 2 = 36÷2=3
Now, 3 is divisible by 3.3÷3=13 \div 3 = 13÷3=1
Thus, the prime factorization of 96 is:96=25×396 = 2^5 \times 396=25×3
Step 2: Prime Factorization of 404
Next, divide 404 by the smallest prime number (2).404÷2=202404 \div 2 = 202404÷2=202202÷2=101202 \div 2 = 101202÷2=101
Since 101 is a prime number, we stop here.
Thus, the prime factorization of 404 is:404=22×101404 = 2^2 \times 101404=22×101
Step 3: Finding the HCF
The HCF is found by taking the lowest power of all the common prime factors.
From the prime factorization, the common prime factor between 96 and 404 is 2.
- The power of 2 in 96 is 252^525.
- The power of 2 in 404 is 222^222.
Thus, the lowest power of 2 is 222^222.
Therefore, the HCF of 96 and 404 is:HCF=22=4HCF = 2^2 = 4HCF=22=4
Step 4: Finding the LCM
The LCM is found by taking the highest power of each prime factor that appears in either of the factorizations.
- The highest power of 2 is 252^525.
- The highest power of 3 is 313^131.
- The highest power of 101 is 1011101^11011.
Therefore, the LCM of 96 and 404 is:LCM=25×31×1011=32×3×101=9696LCM = 2^5 \times 3^1 \times 101^1 = 32 \times 3 \times 101 = 9696LCM=25×31×1011=32×3×101=9696
Final Answer:
- HCF of 96 and 404 = 4
- LCM of 96 and 404 = 9696
Explanation:
The prime factorization method is based on breaking down the numbers into their prime factors and using those factors to find the HCF and LCM. The HCF is determined by multiplying the lowest powers of common prime factors, while the LCM is found by multiplying the highest powers of all prime factors present in either number.
