The LCM of two numbers is 420. Which of the following cannot be the HCF of the two numberS?

Options

A. 70
B. 60
C. 210
D. 80

Answer

To solve the problem, we need to determine which of the given options cannot be the HCF (Highest Common Factor) of two numbers whose LCM (Lowest Common Multiple) is 420.

### Step-by-Step Solution:

1. **Understand the relationship between LCM and HCF**:
The relationship between LCM and HCF of two numbers \( a \) and \( b \) can be expressed as:
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \]
This means that the product of the LCM and HCF of two numbers is equal to the product of the numbers themselves.

2. **Find the prime factorization of 420**:
To find the prime factorization of 420, we can divide it by prime numbers:
- \( 420 \div 2 = 210 \)
- \( 210 \div 2 = 105 \)
- \( 105 \div 3 = 35 \)
- \( 35 \div 5 = 7 \)
- \( 7 \div 7 = 1 \)

Therefore, the prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

3. **Analyze the options for HCF**:
We need to check which of the given options cannot be formed using the prime factors of 420. The options are typically numbers that are factors of 420.

4. **Check each option**:
- **Option A: 70**
- Prime factorization: \( 70 = 2^1 \times 5^1 \times 7^1 \)
- This can be formed using the prime factors of 420.

- **Option B: 60**
- Prime factorization: \( 60 = 2^2 \times 3^1 \times 5^1 \)
- This can also be formed using the prime factors of 420.

- **Option C: 210**
- Prime factorization: \( 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \)
- This can also be formed using the prime factors of 420.

- **Option D: 80**
- Prime factorization: \( 80 = 2^4 \times 5^1 \)
- This cannot be formed using the prime factors of 420 because we only have \( 2^2 \) available.

5. **Conclusion**:
The option that cannot be the HCF of the two numbers is **80**.

### Final Answer:
**80 cannot be the HCF of the two numbers whose LCM is 420.**