Find the HCF of 216, 288 and 720.

Options

A. 72
B. 36
C. 36
D. 82

Answer

To find the HCF (Highest Common Factor) of the numbers 216, 288, and 720, we will follow these steps:

### Step 1: Prime Factorization
First, we need to factor each number into its prime factors.

- **For 216:**
- 216 can be divided by 2:
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27 (27 is not divisible by 2)
- Now, divide 27 by 3:
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 216 is:
\[ 216 = 2^3 \times 3^3 \]

- **For 288:**
- 288 can also be divided by 2:
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9 (9 is not divisible by 2)
- Now, divide 9 by 3:
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 288 is:
\[ 288 = 2^5 \times 3^2 \]

- **For 720:**
- 720 can be divided by 2:
- 720 ÷ 2 = 360
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45 (45 is not divisible by 2)
- Now, divide 45 by 3:
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- So, the prime factorization of 720 is:
\[ 720 = 2^4 \times 3^2 \times 5^1 \]

### Step 2: Identify Common Factors
Next, we identify the common prime factors from the factorizations:

- For 2:
- \(2^3\) (from 216)
- \(2^5\) (from 288)
- \(2^4\) (from 720)
- The minimum power of 2 is \(2^3\).

- For 3:
- \(3^3\) (from 216)
- \(3^2\) (from 288)
- \(3^2\) (from 720)
- The minimum power of 3 is \(3^2\).

- The factor 5 is not common in all three numbers.

### Step 3: Calculate the HCF
Now, we multiply the common prime factors with their minimum powers:

\[ HCF = 2^3 \times 3^2 \]

Calculating this gives:

\[ HCF = 8 \times 9 = 72 \]

### Final Answer
Thus, the HCF of 216, 288, and 720 is **72**.

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