Find the HCF of 120 and 180.

Answer

To find the HCF (Highest Common Factor) of 120 and 180, we can use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 120
1. Start dividing 120 by the smallest prime number, which is 2.
- \( 120 \div 2 = 60 \)
2. Divide 60 by 2 again.
- \( 60 \div 2 = 30 \)
3. Divide 30 by 2 again.
- \( 30 \div 2 = 15 \)
4. Now, 15 is not divisible by 2, so we move to the next prime number, which is 3.
- \( 15 \div 3 = 5 \)
5. Finally, 5 is a prime number itself.
- \( 5 \div 5 = 1 \)

So, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

### Step 2: Prime Factorization of 180
1. Start dividing 180 by 2.
- \( 180 \div 2 = 90 \)
2. Divide 90 by 2 again.
- \( 90 \div 2 = 45 \)
3. Now, 45 is not divisible by 2, so we move to the next prime number, which is 3.
- \( 45 \div 3 = 15 \)
4. Divide 15 by 3 again.
- \( 15 \div 3 = 5 \)
5. Finally, 5 is a prime number itself.
- \( 5 \div 5 = 1 \)

So, the prime factorization of 180 is:
\[ 180 = 2^2 \times 3^2 \times 5^1 \]

### Step 3: Identify Common Factors
Now, we need to identify the common prime factors from both numbers:
- For 2: The minimum power is \( 2^2 \) (from 180).
- For 3: The minimum power is \( 3^1 \) (from 120).
- For 5: The minimum power is \( 5^1 \) (common in both).

### Step 4: Calculate the HCF
Now, we multiply the common factors:
\[ \text{HCF} = 2^2 \times 3^1 \times 5^1 \]
Calculating this:
1. \( 2^2 = 4 \)
2. \( 3^1 = 3 \)
3. \( 5^1 = 5 \)

Now multiply these together:
\[ \text{HCF} = 4 \times 3 \times 5 = 60 \]

Thus, the HCF of 120 and 180 is **60**.

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