Given two numbers 15 and 25, find their HCF and LCM and verify that HCF `xx` LCM `=15 xx 25`

Answer

To solve the problem of finding the HCF and LCM of the numbers 15 and 25, and verifying that HCF × LCM = 15 × 25, we will follow these steps:

### Step 1: Find the Prime Factorization of 15 and 25
- **For 15**:
- 15 can be divided by 3 (15 ÷ 3 = 5).
- So, the prime factorization of 15 is \(3 \times 5\).

- **For 25**:
- 25 can be divided by 5 (25 ÷ 5 = 5).
- So, the prime factorization of 25 is \(5 \times 5\) or \(5^2\).

### Step 2: Identify the HCF (Highest Common Factor)
- The common prime factor between 15 and 25 is 5.
- Therefore, the HCF of 15 and 25 is **5**.

### Step 3: Identify the LCM (Lowest Common Multiple)
- To find the LCM, we take the highest power of each prime factor present in the factorizations:
- For 3: The highest power is \(3^1\) (from 15).
- For 5: The highest power is \(5^2\) (from 25).

- Thus, the LCM is calculated as:
\[ \text{LCM} = 3^1 \times 5^2 = 3 \times 25 = 75. \]

### Step 4: Verify the relationship HCF × LCM = 15 × 25
- Now we calculate HCF × LCM:
\[ \text{HCF} \times \text{LCM} = 5 \times 75 = 375. \]

- Next, we calculate 15 × 25:
\[ 15 \times 25 = 375. \]

- Since both calculations yield the same result, we have verified that:
\[ \text{HCF} \times \text{LCM} = 15 \times 25. \]

### Final Answer:
- HCF of 15 and 25 is **5**.
- LCM of 15 and 25 is **75**.
- Verification: \(5 \times 75 = 15 \times 25 = 375\).

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