Find the HCF of 120, 180 and 250 by prime factorisation method.

Answer

To find the HCF (Highest Common Factor) of 120, 180, and 250 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number

**For 120:**
1. Divide by 2: \(120 \div 2 = 60\)
2. Divide by 2: \(60 \div 2 = 30\)
3. Divide by 2: \(30 \div 2 = 15\)
4. Divide by 3: \(15 \div 3 = 5\)
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 180:**
1. Divide by 2: \(180 \div 2 = 90\)
2. Divide by 2: \(90 \div 2 = 45\)
3. Divide by 3: \(45 \div 3 = 15\)
4. Divide by 3: \(15 \div 3 = 5\)
5. Divide by 5: \(5 \div 5 = 1\)

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

**For 250:**
1. Divide by 2: \(250 \div 2 = 125\)
2. Divide by 5: \(125 \div 5 = 25\)
3. Divide by 5: \(25 \div 5 = 5\)
4. Divide by 5: \(5 \div 5 = 1\)

So, the prime factorization of 250 is:
\[ 250 = 2^1 \times 5^3 \]

### Step 2: Identify Common Prime Factors

Now, we will list the prime factors of each number:

- **120:** \(2^3, 3^1, 5^1\)
- **180:** \(2^2, 3^2, 5^1\)
- **250:** \(2^1, 5^3\)

### Step 3: Find the Lowest Power of Common Prime Factors

Now, we identify the common prime factors and their lowest powers:

- For \(2\): The lowest power is \(2^1\) (from 250).
- For \(3\): It is not common in all three numbers.
- For \(5\): The lowest power is \(5^1\) (common in all).

### Step 4: Calculate the HCF

Now we multiply the common prime factors with their lowest powers:

\[ \text{HCF} = 2^1 \times 5^1 = 2 \times 5 = 10 \]

### Conclusion

Thus, the HCF of 120, 180, and 250 is **10**.

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