Three measuring rods are 45 cm, 50 cm and 75 cm in length. What is the least length of a rope that can be measured by the full length of each of these three rods ?

Answer

To find the least length of a rope that can be measured by the full length of each of the three measuring rods (45 cm, 50 cm, and 75 cm), we need to calculate the least common multiple (LCM) of these three lengths. Here’s how we can do it step by step:

### Step 1: List the lengths of the rods
The lengths of the rods are:
- Rod 1: 45 cm
- Rod 2: 50 cm
- Rod 3: 75 cm

### Step 2: Prime factorization of each length
We will find the prime factorization of each length:

- **45**:
- 45 = 3 × 15
- 15 = 3 × 5
- So, 45 = 3² × 5¹

- **50**:
- 50 = 2 × 25
- 25 = 5 × 5
- So, 50 = 2¹ × 5²

- **75**:
- 75 = 3 × 25
- 25 = 5 × 5
- So, 75 = 3¹ × 5²

### Step 3: Identify the highest powers of each prime factor
Now we will identify the highest power of each prime factor from the factorizations:

- For the prime number **2**: The highest power is 2¹ (from 50).
- For the prime number **3**: The highest power is 3² (from 45).
- For the prime number **5**: The highest power is 5² (from both 50 and 75).

### Step 4: Calculate the LCM
Now we can calculate the LCM by multiplying the highest powers of all prime factors together:

LCM = 2¹ × 3² × 5²
= 2 × 9 × 25
= 2 × 225
= 450

### Conclusion
The least length of rope that can be measured by the full length of each of these three rods is **450 cm**.

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