HCF of 135 and 225 by Prime Factorization

To find the HCF (Highest Common Factor) of 135 and 225 using prime factorization, we first need to determine the prime factors of each number.

Step 1: Prime Factorization of 135

• 135 is divisible by 3: 135 ÷ 3 = 45
• 45 is also divisible by 3: 45 ÷ 3 = 15
• 15 is divisible by 3: 15 ÷ 3 = 5
• 5 is a prime number.

So, the prime factorization of 135 is: 3 × 3 × 3 × 5 or 3³ × 5.

Step 2: Prime Factorization of 225

• 225 is divisible by 3: 225 ÷ 3 = 75
• 75 is also divisible by 3: 75 ÷ 3 = 25
• 25 is divisible by 5: 25 ÷ 5 = 5
• 5 is a prime number.

So, the prime factorization of 225 is: 3 × 3 × 5 × 5 or 3² × 5².

Step 3: Identifying the Common Factors

Now that we have the prime factors:

• 135: 3³ × 5
• 225: 3² × 5²

The common prime factors are 3 and 5. To find the HCF, we take the lowest power of these common factors:

• For 3, the lowest power is 3².
• For 5, the lowest power is 5¹.

Step 4: Calculating the HCF

Now we multiply these together:

HCF = 3² × 5¹ = 9 × 5 = 45

Conclusion:

Thus, the HCF of 135 and 225 by prime factorization is 45.