Find the HCF and LCM of 45 and 99. Verify that `45 xx 99= HCF xx LCM`

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of 45 and 99, and to verify that \( 45 \times 99 = \text{HCF} \times \text{LCM} \), we can follow these steps:

### Step 1: Prime Factorization of 45
1. Start with the number 45.
2. The smallest prime number that divides 45 is 3.
- \( 45 \div 3 = 15 \)
3. Next, divide 15 by 3.
- \( 15 \div 3 = 5 \)
4. Now, 5 is a prime number, so we stop here.
5. The prime factorization of 45 is:
\[ 45 = 3^2 \times 5^1 \]

### Step 2: Prime Factorization of 99
1. Start with the number 99.
2. The smallest prime number that divides 99 is 3.
- \( 99 \div 3 = 33 \)
3. Next, divide 33 by 3.
- \( 33 \div 3 = 11 \)
4. Now, 11 is a prime number, so we stop here.
5. The prime factorization of 99 is:
\[ 99 = 3^2 \times 11^1 \]

### Step 3: Finding the HCF
1. Identify the common prime factors from the factorizations of 45 and 99.
- The common prime factor is \( 3 \).
2. Take the lowest power of the common prime factor:
- \( 3^2 \) (from both factorizations).
3. Therefore, the HCF is:
\[ \text{HCF} = 3^2 = 9 \]

### Step 4: Finding the LCM
1. For the LCM, take all the prime factors from both numbers, using the highest power for each:
- From 45: \( 3^2 \) and \( 5^1 \)
- From 99: \( 3^2 \) and \( 11^1 \)
2. The LCM is calculated as:
\[ \text{LCM} = 3^2 \times 5^1 \times 11^1 \]
3. Calculate:
- \( 3^2 = 9 \)
- \( 5^1 = 5 \)
- \( 11^1 = 11 \)
- Now multiply:
\[ 9 \times 5 = 45 \]
\[ 45 \times 11 = 495 \]
4. Therefore, the LCM is:
\[ \text{LCM} = 495 \]

### Step 5: Verification
1. Calculate \( 45 \times 99 \):
\[ 45 \times 99 = 4455 \]
2. Calculate \( \text{HCF} \times \text{LCM} \):
\[ 9 \times 495 = 4455 \]
3. Since both products are equal:
\[ 45 \times 99 = \text{HCF} \times \text{LCM} \]

### Final Answer
- HCF of 45 and 99 is **9**.
- LCM of 45 and 99 is **495**.
- Verification: \( 45 \times 99 = 9 \times 495 \).