Three wooden blocks are of size 36 m, 54 m & 63 m respectively. They are to be divided into smaller blocks of equal size. What would be the size of smaller block ?

Options

A. 9 m.
B. 18 m.
C. 51 m.
D. 4.5 m.

Answer

To find the size of the smaller block that can be obtained from the three wooden blocks of sizes 36 m, 54 m, and 63 m, we need to calculate the Highest Common Factor (HCF) of these three numbers. Here’s how to do it step by step:

### Step 1: Prime Factorization
First, we will perform the prime factorization of each of the three numbers.

1. **For 36:**
- 36 can be divided by 2:
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

2. **For 54:**
- 54 can be divided by 2:
- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 54 is:
\[ 54 = 2^1 \times 3^3 \]

3. **For 63:**
- 63 can be divided by 3:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- So, the prime factorization of 63 is:
\[ 63 = 3^2 \times 7^1 \]

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

- The common prime factor among 36, 54, and 63 is **3**.
- The powers of 3 in the factorizations are:
- For 36: \(3^2\)
- For 54: \(3^3\)
- For 63: \(3^2\)

The lowest power of 3 common to all three numbers is \(3^2\).

### Step 3: Calculate HCF
Now we can calculate the HCF by multiplying the common prime factors with their lowest powers:

\[ \text{HCF} = 3^2 = 9 \]

### Conclusion
Thus, the size of the smaller block that can be obtained from the three wooden blocks is **9 meters**.

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