Find the greatest number which on dividing 1580 and 3800 leaves remainders 8 and 1 respectively.

Options

A. A. 262
B. B. 131
C. C. 65.5
D. D. 393

Answer

To find the greatest number which, when dividing 1580 and 3800, leaves remainders of 8 and 1 respectively, we can follow these steps:

### Step 1: Set Up the Problem
We need to find a number \( N \) such that:
- \( 1580 \mod N = 8 \)
- \( 3800 \mod N = 1 \)

This means that:
- \( 1580 - 8 = 1572 \) must be divisible by \( N \)
- \( 3800 - 1 = 3799 \) must be divisible by \( N \)

### Step 2: Rewrite the Problem
We need to find \( N \) such that:
- \( N \) divides both \( 1572 \) and \( 3799 \).

### Step 3: Calculate the HCF
To find \( N \), we need to calculate the Highest Common Factor (HCF) of \( 1572 \) and \( 3799 \).

### Step 4: Factor the Numbers
1. **Factor \( 1572 \)**:
- \( 1572 = 2 \times 786 \)
- \( 786 = 2 \times 393 \)
- \( 393 = 3 \times 131 \)
- So, \( 1572 = 2^2 \times 3 \times 131 \).

2. **Factor \( 3799 \)**:
- \( 3799 = 29 \times 131 \).

### Step 5: Find the HCF
The common factor between \( 1572 \) and \( 3799 \) is \( 131 \).

### Step 6: Conclusion
Thus, the greatest number \( N \) that divides both \( 1572 \) and \( 3799 \) is \( 131 \).

### Final Answer
The greatest number is **131**.

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