The sides of a triangular piece of ground measure 15547 , 17647, 3521 metres respectively . Find the length of the largest hurdle that can be used to fence it exactly without bending or cutting a hurdle .

Options

A. 6 m
B. `6.5` m
C. 7 m
D. `7.5` m

Answer

To find the length of the largest hurdle that can be used to fence a triangular piece of ground with sides measuring 15547 m, 17647 m, and 3521 m, we need to determine the highest common factor (HCF) of these three numbers. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
The sides of the triangular piece of ground are:
- Side 1: 15547 m
- Side 2: 17647 m
- Side 3: 3521 m

### Step 2: Find the prime factorization of each side
We will factor each of these numbers to find their prime factors.

1. **Factor 15547:**
- Divide by 7:
- 15547 ÷ 7 = 2221
- Now, we need to factor 2221. After checking for divisibility, we find that:
- 2221 is a prime number.
- So, the factorization of 15547 is:
- \( 15547 = 7 \times 2221 \)

2. **Factor 17647:**
- Divide by 7:
- 17647 ÷ 7 = 2521
- Now, we need to factor 2521. After checking for divisibility, we find that:
- 2521 is also a prime number.
- So, the factorization of 17647 is:
- \( 17647 = 7 \times 2521 \)

3. **Factor 3521:**
- Divide by 7:
- 3521 ÷ 7 = 503
- Now, we need to factor 503. After checking for divisibility, we find that:
- 503 is a prime number.
- So, the factorization of 3521 is:
- \( 3521 = 7 \times 503 \)

### Step 3: Identify the common factors
Now that we have the prime factorizations:
- \( 15547 = 7 \times 2221 \)
- \( 17647 = 7 \times 2521 \)
- \( 3521 = 7 \times 503 \)

The common factor in all three numbers is **7**.

### Step 4: Conclusion
The largest hurdle that can be used to fence the triangular piece of ground exactly without bending or cutting is the HCF of the three sides, which is:
- **HCF = 7 meters**

Thus, the length of the largest hurdle is **7 meters**.

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