The HCF of 135 and 225 is:

Options

A. 15
B. 75
C. 45
D. 5

Answer

To find the HCF (Highest Common Factor) of 135 and 225 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 135
1. Start by dividing 135 by the smallest prime number, which is 3.
- \( 135 \div 3 = 45 \)
2. Next, divide 45 by 3.
- \( 45 \div 3 = 15 \)
3. Then, divide 15 by 3.
- \( 15 \div 3 = 5 \)
4. Finally, 5 is a prime number and cannot be divided further.

So, the prime factorization of 135 is:
\[ 135 = 3^3 \times 5^1 \]

### Step 2: Prime Factorization of 225
1. Start by dividing 225 by the smallest prime number, which is 3.
- \( 225 \div 3 = 75 \)
2. Next, divide 75 by 3.
- \( 75 \div 3 = 25 \)
3. Then, divide 25 by 5.
- \( 25 \div 5 = 5 \)
4. Finally, 5 is a prime number and cannot be divided further.

So, the prime factorization of 225 is:
\[ 225 = 3^2 \times 5^2 \]

### Step 3: Identify Common Factors
Now, we need to identify the common prime factors from both factorizations:
- For 3: The minimum power is \( 3^2 \) (from 225).
- For 5: The minimum power is \( 5^1 \) (from 135).

### Step 4: Calculate HCF
Now, we multiply the common prime factors:
\[ \text{HCF} = 3^2 \times 5^1 = 9 \times 5 = 45 \]

Thus, the HCF of 135 and 225 is **45**.

### Summary of the Steps
1. Factor 135 into primes: \( 3^3 \times 5^1 \).
2. Factor 225 into primes: \( 3^2 \times 5^2 \).
3. Identify common factors: \( 3^2 \) and \( 5^1 \).
4. Multiply the common factors to find HCF: \( 3^2 \times 5^1 = 45 \).