Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the same distance in complete steps ?

Answer

To find the minimum distance that each boy should cover so that they all can take complete steps, we need to determine the Least Common Multiple (LCM) of their step lengths: 63 cm, 70 cm, and 77 cm.

### Step-by-Step Solution:

1. **Identify the Step Lengths**:
- The step lengths of the three boys are:
- Boy 1: 63 cm
- Boy 2: 70 cm
- Boy 3: 77 cm

2. **Find the Prime Factorization of Each Step Length**:
- For 63:
- 63 = 3 × 3 × 7 = \(3^2 \times 7^1\)
- For 70:
- 70 = 2 × 5 × 7 = \(2^1 \times 5^1 \times 7^1\)
- For 77:
- 77 = 7 × 11 = \(7^1 \times 11^1\)

3. **List the Prime Factors**:
- The prime factors we have are: 2, 3, 5, 7, and 11.

4. **Determine the Highest Power of Each Prime Factor**:
- From the factorizations:
- \(2^1\) (from 70)
- \(3^2\) (from 63)
- \(5^1\) (from 70)
- \(7^1\) (from all three)
- \(11^1\) (from 77)

5. **Calculate the LCM**:
- LCM = \(2^1 \times 3^2 \times 5^1 \times 7^1 \times 11^1\)
- Calculate:
- \(2^1 = 2\)
- \(3^2 = 9\)
- \(5^1 = 5\)
- \(7^1 = 7\)
- \(11^1 = 11\)
- Now multiply these together:
- \(2 \times 9 = 18\)
- \(18 \times 5 = 90\)
- \(90 \times 7 = 630\)
- \(630 \times 11 = 6930\)

6. **Conclusion**:
- The LCM of 63, 70, and 77 is 6930 cm.
- Therefore, the minimum distance each boy should cover so that all can take complete steps is **6930 cm**, which can also be expressed as **6.93 kilometers**.