11. Find LCM and HCF of 14 and 40. Verify that product numbers is equal to the product their LCM and HCF.

Let's solve this step by step.

### Step 1: Find the Highest Common Factor (HCF) of 14 and 40
- **Prime Factorization of 14**: \( 14 = 2 \times 7 \)
- **Prime Factorization of 40**: \( 40 = 2^3 \times 5 \)

The common factor in both numbers is 2.

So, the **HCF** is \( 2 \).

### Step 2: Find the Lowest Common Multiple (LCM) of 14 and 40
- List all prime factors of both numbers, taking the highest power of each factor:
  - For 14: \( 2 \times 7 \)
  - For 40: \( 2^3 \times 5 \)

The LCM is obtained by multiplying the highest powers of all prime factors:

\[
\text{LCM} = 2^3 \times 5 \times 7 = 8 \times 5 \times 7 = 280
\]

So, the **LCM** is \( 280 \).

### Step 3: Verify the Product of Numbers Equals the Product of Their LCM and HCF

- **Product of the numbers 14 and 40**: 
  \[
  14 \times 40 = 560
  \]

- **Product of the LCM and HCF**:
  \[
  280 \times 2 = 560
  \]

Since both products are equal, the calculation is verified.

### Summary:
- **HCF** of 14 and 40: \( 2 \)
- **LCM** of 14 and 40: \( 280 \)
- Verification: \( 14 \times 40 = 2 \times 280 = 560 \)

Would you like more details or have any questions?

### Related Questions:
1. Find the HCF and LCM of 18 and 24. Verify the result.
2. Explain why the product of HCF and LCM equals the product of the two numbers.
3. Determine the HCF and LCM of 15 and 35.
4. Find the HCF of three numbers: 14, 28, and 42.
5. Calculate the LCM of 25, 30, and 35.

**Tip**: To verify the correctness of HCF and LCM, always check if the product of the HCF and LCM equals the product of the original numbers.

Learn how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the numbers 14 and 40 using prime factorization. Verify that the product of the numbers equals the product of their LCM and HCF. Suitable for students in Grades 6-8 studying number theory and arithmetic.

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