The length of three rods is 64 cm., 80 cm. and 96 cm. Find the least length of a cloth which can be measured by using any of the three rods.

Options

A. 0.96 m.
B. 19.20 m.
C. 9.60 m.
D. 96 m.

Answer

To find the least length of cloth that can be measured using any of the three rods with lengths 64 cm, 80 cm, and 96 cm, we need to calculate the Least Common Multiple (LCM) of these three lengths. Here’s how to do it step by step:

### Step 1: Prime Factorization of Each Length
- **64 cm**:
- 64 = 2 × 32
- 32 = 2 × 16
- 16 = 2 × 8
- 8 = 2 × 4
- 4 = 2 × 2
- Therefore, the prime factorization of 64 is \( 2^6 \).

- **80 cm**:
- 80 = 8 × 10
- 8 = 2 × 4
- 4 = 2 × 2
- 10 = 2 × 5
- Therefore, the prime factorization of 80 is \( 2^4 \times 5^1 \).

- **96 cm**:
- 96 = 12 × 8
- 12 = 3 × 4
- 4 = 2 × 2
- 8 = 2 × 4
- Therefore, the prime factorization of 96 is \( 2^5 \times 3^1 \).

### Step 2: Identify the Highest Powers of Each Prime Factor
Now we will take the highest power of each prime factor from the factorizations:
- For \( 2 \): The highest power is \( 2^6 \) (from 64).
- For \( 3 \): The highest power is \( 3^1 \) (from 96).
- For \( 5 \): The highest power is \( 5^1 \) (from 80).

### Step 3: Calculate the LCM
Now we can calculate the LCM by multiplying the highest powers of all prime factors:
\[ \text{LCM} = 2^6 \times 3^1 \times 5^1 \]
Calculating this step by step:
- \( 2^6 = 64 \)
- \( 3^1 = 3 \)
- \( 5^1 = 5 \)

Now multiply them together:
\[ \text{LCM} = 64 \times 3 \times 5 \]
Calculating \( 64 \times 3 = 192 \) and then \( 192 \times 5 = 960 \).

### Final Answer
The least length of cloth that can be measured using any of the three rods is **960 cm**.

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