Find the LCM and HCF of 120 and 84 and verify its relation between the two numbers 

To find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of 120 and 84, and to verify the relation between them, we will follow these steps:

### Step 1: Prime Factorization of 120 and 84

1. **Prime factorization of 120**:
   \[
   120 = 2^3 \times 3 \times 5
   \]

2. **Prime factorization of 84**:
   \[
   84 = 2^2 \times 3 \times 7
   \]

### Step 2: Finding the HCF (Highest Common Factor)
The HCF is obtained by taking the lowest powers of all common prime factors.

- Common prime factors: \( 2 \) and \( 3 \)
- Lowest power of \( 2 \): \( 2^2 \)
- Lowest power of \( 3 \): \( 3^1 \)

Thus, the HCF is:
\[
HCF = 2^2 \times 3 = 4 \times 3 = 12
\]

### Step 3: Finding the LCM (Least Common Multiple)
The LCM is obtained by taking the highest powers of all prime factors present in both numbers.

- Prime factors: \( 2, 3, 5, 7 \)
- Highest power of \( 2 \): \( 2^3 \)
- Highest power of \( 3 \): \( 3^1 \)
- Highest power of \( 5 \): \( 5^1 \)
- Highest power of \( 7 \): \( 7^1 \)

Thus, the LCM is:
\[
LCM = 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 = 840
\]

### Step 4: Verifying the Relationship Between HCF and LCM
The relationship between the HCF and LCM of two numbers is given by:
\[
\text{HCF} \times \text{LCM} = \text{Product of the two numbers}
\]
For 120 and 84:
\[
\text{HCF} \times \text{LCM} = 12 \times 840 = 10080
\]
And the product of the two numbers is:
\[
120 \times 84 = 10080
\]
Since both sides are equal, the relationship is verified.

### Summary:
- **HCF**: 12
- **LCM**: 840
- **Verification**: \( \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \)

Would you like further details on any step? Here are some related questions you might consider:

1. What is the process for finding the HCF using the Euclidean algorithm?
2. How does the relationship between HCF and LCM help in solving other problems?
3. Can we use HCF and LCM to simplify fractions?
4. How do we find the LCM of more than two numbers?
5. What are some real-life applications of HCF and LCM?

**Tip**: Remember, the product of the HCF and LCM of two numbers is always equal to the product of the two numbers.

Find the LCM and HCF of 120 and 84 and verify its relation between the two numbers.

Learn how to find the Least Common Multiple (LCM) and Highest Common Factor (HCF) of 120 and 84 using prime factorization. This solution explains the relation between LCM, HCF, and the product of two numbers, suitable for Grades 6-8.

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