During a morning walk, four boys step off together. Their steps measure 70 cm, 65 cm, 75 cm and 80 cm respectively. At what distance from the starting point will they step off together again ?

Options

A. 1000 m
B. 1092 m
C. 980 m
D. 900 m

Answer

To find the distance at which the four boys will step off together again, we need to calculate the Least Common Multiple (LCM) of their step lengths. The step lengths are 70 cm, 65 cm, 75 cm, and 80 cm.

### Step-by-Step Solution:

1. **List the step lengths**:
The step lengths are: 70 cm, 65 cm, 75 cm, and 80 cm.

2. **Prime Factorization**:
- **70**: \( 70 = 2 \times 5 \times 7 \)
- **65**: \( 65 = 5 \times 13 \)
- **75**: \( 75 = 3 \times 5^2 \)
- **80**: \( 80 = 2^4 \times 5 \)

3. **Identify the highest powers of each prime factor**:
- For \(2\): The highest power is \(2^4\) (from 80).
- For \(3\): The highest power is \(3^1\) (from 75).
- For \(5\): The highest power is \(5^2\) (from 75).
- For \(7\): The highest power is \(7^1\) (from 70).
- For \(13\): The highest power is \(13^1\) (from 65).

4. **Calculate the LCM**:
LCM = \(2^4 \times 3^1 \times 5^2 \times 7^1 \times 13^1\)

Now, let's calculate it step by step:
- \(2^4 = 16\)
- \(3^1 = 3\)
- \(5^2 = 25\)
- \(7^1 = 7\)
- \(13^1 = 13\)

Now multiply these together:
- \(16 \times 3 = 48\)
- \(48 \times 25 = 1200\)
- \(1200 \times 7 = 8400\)
- \(8400 \times 13 = 109200\)

So, the LCM is \(109200\) cm.

5. **Convert to meters**:
Since \(1\) meter = \(100\) cm, we convert \(109200\) cm to meters:
\[ \frac{109200 \text{ cm}}{100} = 1092 \text{ meters} \]

Thus, the distance from the starting point where they will step off together again is **1092 meters**.