Find the HCF of 120 and 300 using prime factors.

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To find the Highest Common Factor (HCF) of 120 and 300 using prime factors, we will follow these steps:

1. **Find the prime factorization of each number.**
2. **Identify the common prime factors.**
3. **Multiply the common prime factors to get the HCF.**

### Step 1: Prime Factorization

**For 120:**
- Divide by 2: $$ 120 \div 2 = 60 $$
- Divide by 2: $$ 60 \div 2 = 30 $$
- Divide by 2: $$ 30 \div 2 = 15 $$
- Divide by 3: $$ 15 \div 3 = 5 $$
- Divide by 5: $$ 5 \div 5 = 1 $$

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

**For 300:**
- Divide by 2: $$ 300 \div 2 = 150 $$
- Divide by 2: $$ 150 \div 2 = 75 $$
- Divide by 3: $$ 75 \div 3 = 25 $$
- Divide by 5: $$ 25 \div 5 = 5 $$
- Divide by 5: $$ 5 \div 5 = 1 $$

So, the prime factorization of 300 is:
$$
300 = 2^2 \times 3^1 \times 5^2
$$

### Step 2: Identify Common Prime Factors

Now, we will identify the common prime factors from both factorizations:
- For $$2$$: The minimum power is $$2^2$$.
- For $$3$$: The minimum power is $$3^1$$.
- For $$5$$: The minimum power is $$5^1$$.

### Step 3: Multiply the Common Prime Factors

Now, we multiply the common prime factors:
$$
\text{HCF} = 2^2 \times 3^1 \times 5^1
$$

Calculating this:
$$
\text{HCF} = 4 \times 3 \times 5
$$
$$
= 12 \times 5 = 60
$$

Thus, the HCF of 120 and 300 is $$ \boxed{60} $$.
The HCF of 120 and 300 is 60.

Find the HCF of 120 and 300 using prime factors.

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