In order to find out the H.C.F. of two numbers by division method. It was found that the final divisor is 37 and successive quotients are 7, 2, 3 respectively. Find both no's.

Options

A. 259, 1924
B. 259, 1724
C. 259, 1942
D. 295, 1924

Answer

To find the two numbers using the division method and the given information about the HCF, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Division Method**:
The division method for finding HCF involves repeatedly dividing the larger number by the smaller number and using the remainder until we reach a remainder of 0. The last non-zero divisor is the HCF.

2. **Setting Up the Division**:
We know that the final divisor (HCF) is 37, and the successive quotients are 7, 2, and 3. We can set up the division as follows:
- Let the first number be \( n_1 \) and the second number be \( n_2 \).
- The division can be represented as:
- \( n_2 = 37 \times 3 + r_1 \) (where \( r_1 \) is the remainder)
- \( r_1 = 37 \times 2 + r_2 \)
- \( r_2 = 37 \times 7 + 0 \)

3. **Calculating the Remainders**:
- From the last division, we know that:
\[ r_2 = 37 \times 3 = 111 \]
- Now substituting \( r_2 \) into the second equation:
\[ r_1 = 111 \times 2 + 37 = 222 + 37 = 259 \]

4. **Finding the First Number \( n_1 \)**:
- We have found \( n_1 = 259 \).

5. **Finding the Second Number \( n_2 \)**:
- Now, we can find \( n_2 \) using the first quotient:
\[ n_2 = 259 \times 7 + 111 \]
- Calculating \( n_2 \):
\[ n_2 = 1813 + 111 = 1924 \]

6. **Final Result**:
- The two numbers are \( n_1 = 259 \) and \( n_2 = 1924 \).

### Final Answer:
The two numbers are **259 and 1924**.

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