The largest number which exactly divides 210, 315, 147 and 168 is -

Options

A. 3
B. 7
C. 21
D. 4410

Answer

To find the largest number that exactly divides 210, 315, 147, and 168, we will calculate the Highest Common Factor (HCF) using the prime factorization method. Here are the steps:

### Step 1: Prime Factorization of Each Number
1. **Factor 210**:
- 210 can be divided by 2: \( 210 \div 2 = 105 \)
- 105 can be divided by 3: \( 105 \div 3 = 35 \)
- 35 can be divided by 5: \( 35 \div 5 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

2. **Factor 315**:
- 315 can be divided by 3: \( 315 \div 3 = 105 \)
- 105 can be divided by 3 again: \( 105 \div 3 = 35 \)
- 35 can be divided by 5: \( 35 \div 5 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 315 is:
\[ 315 = 3^2 \times 5^1 \times 7^1 \]

3. **Factor 147**:
- 147 can be divided by 3: \( 147 \div 3 = 49 \)
- 49 can be divided by 7: \( 49 \div 7 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 147 is:
\[ 147 = 3^1 \times 7^2 \]

4. **Factor 168**:
- 168 can be divided by 2: \( 168 \div 2 = 84 \)
- 84 can be divided by 2: \( 84 \div 2 = 42 \)
- 42 can be divided by 2: \( 42 \div 2 = 21 \)
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 168 is:
\[ 168 = 2^3 \times 3^1 \times 7^1 \]

### Step 2: Identify Common Factors
Now, we will identify the common prime factors from the factorizations we obtained:
- **For 2**:
- 210: \( 2^1 \)
- 315: \( 2^0 \) (not present)
- 147: \( 2^0 \) (not present)
- 168: \( 2^3 \)
- **Common factor**: \( 2^0 \) (not included in HCF)

- **For 3**:
- 210: \( 3^1 \)
- 315: \( 3^2 \)
- 147: \( 3^1 \)
- 168: \( 3^1 \)
- **Common factor**: \( 3^1 \)

- **For 5**:
- 210: \( 5^1 \)
- 315: \( 5^1 \)
- 147: \( 5^0 \) (not present)
- 168: \( 5^0 \) (not present)
- **Common factor**: \( 5^0 \) (not included in HCF)

- **For 7**:
- 210: \( 7^1 \)
- 315: \( 7^1 \)
- 147: \( 7^2 \)
- 168: \( 7^1 \)
- **Common factor**: \( 7^1 \)

### Step 3: Calculate HCF
Now we multiply the common factors:
\[ \text{HCF} = 3^1 \times 7^1 = 3 \times 7 = 21 \]

### Final Answer
The largest number that exactly divides 210, 315, 147, and 168 is **21**.

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