Find the HCF of 72, 288 and 1080.

Answer

To find the HCF (Highest Common Factor) of the numbers 72, 288, and 1080, we can follow these steps:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 72:**
- 72 can be divided by 2:
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

2. **Prime Factorization of 288:**
- 288 can be divided by 2:
- \( 288 \div 2 = 144 \)
- \( 144 \div 2 = 72 \)
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

3. **Prime Factorization of 1080:**
- 1080 can be divided by 2:
- \( 1080 \div 2 = 540 \)
- \( 540 \div 2 = 270 \)
- \( 270 \div 2 = 135 \)
- \( 135 \div 3 = 45 \)
- \( 45 \div 3 = 15 \)
- \( 15 \div 3 = 5 \)
- \( 5 \div 5 = 1 \)
- So, the prime factorization of 1080 is:
\[ 1080 = 2^3 \times 3^3 \times 5^1 \]

### Step 2: Identify the Common Factors

Now, we will identify the common prime factors from the factorizations:

- For \( 2 \):
- In 72: \( 2^3 \)
- In 288: \( 2^5 \)
- In 1080: \( 2^3 \)
- The minimum power of 2 is \( 2^3 \).

- For \( 3 \):
- In 72: \( 3^2 \)
- In 288: \( 3^2 \)
- In 1080: \( 3^3 \)
- The minimum power of 3 is \( 3^2 \).

- For \( 5 \):
- In 72: \( 5^0 \) (not present)
- In 288: \( 5^0 \) (not present)
- In 1080: \( 5^1 \)
- The minimum power of 5 is \( 5^0 \) (not included).

### Step 3: Calculate the HCF

Now, we can calculate the HCF by multiplying the common prime factors with their minimum powers:
\[ \text{HCF} = 2^3 \times 3^2 = 8 \times 9 = 72 \]

### Final Answer
The HCF of 72, 288, and 1080 is **72**.

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