Find the HCF of he numbers in each of the following , using the prime factorization method :
84 , 98

Options

A. 13
B. 12
C. 15
D. 14

Answer

To find the HCF (Highest Common Factor) of the numbers 84 and 98 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 84
1. Start dividing 84 by the smallest prime number, which is 2.
- \( 84 \div 2 = 42 \)
2. Divide 42 by 2 again.
- \( 42 \div 2 = 21 \)
3. Now, divide 21 by the next smallest prime number, which is 3.
- \( 21 \div 3 = 7 \)
4. Finally, divide 7 by itself (the last prime number).
- \( 7 \div 7 = 1 \)

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

### Step 2: Prime Factorization of 98
1. Start dividing 98 by the smallest prime number, which is 2.
- \( 98 \div 2 = 49 \)
2. Now, divide 49 by the next smallest prime number, which is 7.
- \( 49 \div 7 = 7 \)
3. Finally, divide 7 by itself.
- \( 7 \div 7 = 1 \)

So, the prime factorization of 98 is:
\[ 98 = 2^1 \times 7^2 \]

### Step 3: Identify Common Prime Factors
Now, we will find the common prime factors from both factorizations:
- For 84: \( 2^2 \times 3^1 \times 7^1 \)
- For 98: \( 2^1 \times 7^2 \)

The common prime factors are:
- \( 2^1 \) (the minimum power of 2)
- \( 7^1 \) (the minimum power of 7)

### Step 4: Calculate the HCF
Now, we multiply the common prime factors:
\[ HCF = 2^1 \times 7^1 = 2 \times 7 = 14 \]

Thus, the HCF of 84 and 98 is **14**.

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