What is the HCF of 21, 84 and 49 ?

Options

A. A)1
B. B)7
C. C)21
D. D)3

Answer

To find the HCF (Highest Common Factor) of the numbers 21, 84, and 49, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 21**:
- 21 can be divided by 3 (21 ÷ 3 = 7).
- Thus, the prime factorization of 21 is:
\[ 21 = 3 \times 7 \]

- **For 84**:
- 84 can be divided by 2 (84 ÷ 2 = 42).
- 42 can be divided by 2 again (42 ÷ 2 = 21).
- Now, we factor 21 (as done above): 21 = 3 × 7.
- Thus, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3 \times 7 \]

- **For 49**:
- 49 can be divided by 7 (49 ÷ 7 = 7).
- Thus, the prime factorization of 49 is:
\[ 49 = 7^2 \]

### Step 2: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:

- The prime factorization of 21 is \(3 \times 7\).
- The prime factorization of 84 is \(2^2 \times 3 \times 7\).
- The prime factorization of 49 is \(7^2\).

The common prime factor among all three numbers is **7**.

### Step 3: Determine the HCF
The HCF is determined by taking the lowest power of the common prime factors:

- The only common prime factor is 7, and its lowest power in the factorizations is \(7^1\).

Thus, the HCF of 21, 84, and 49 is:
\[ \text{HCF} = 7 \]

### Final Answer
The HCF of 21, 84, and 49 is **7**.

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