What is least possible number when it is divided by 13 leaves 8 and when divided by 7 leaves remainder 6?

Options

A. 38
B. 34
C. 40
D. 41

Answer

To find the least possible number that leaves a remainder of 8 when divided by 13 and a remainder of 6 when divided by 7, we can set up the problem using modular arithmetic.

### Step-by-Step Solution:

1. **Set Up the Equations**:
We need to express the conditions given in the problem using equations:
- The first condition states that when the number \( x \) is divided by 13, it leaves a remainder of 8:
\[ x \equiv 8 \ (\text{mod} \ 13) \]
- The second condition states that when the number \( x \) is divided by 7, it leaves a remainder of 6:
\[ x \equiv 6 \ (\text{mod} \ 7) \]

2. **Express the First Condition**:
From the first equation, we can express \( x \) in terms of 13:
\[ x = 13k + 8 \quad \text{for some integer } k \]

3. **Substitute into the Second Condition**:
Now, substitute \( x \) from the first equation into the second condition:
\[ 13k + 8 \equiv 6 \ (\text{mod} \ 7) \]
Simplifying this:
\[ 13k \equiv 6 - 8 \ (\text{mod} \ 7) \]
\[ 13k \equiv -2 \ (\text{mod} \ 7) \]
Since \( 13 \equiv 6 \ (\text{mod} \ 7) \), we can replace 13:
\[ 6k \equiv -2 \ (\text{mod} \ 7) \]
To simplify \(-2\) in mod 7, we can add 7:
\[ 6k \equiv 5 \ (\text{mod} \ 7) \]

4. **Find the Inverse of 6 Modulo 7**:
We need to find the multiplicative inverse of 6 modulo 7. The inverse is a number \( m \) such that:
\[ 6m \equiv 1 \ (\text{mod} \ 7) \]
Testing values, we find that \( m = 6 \) works since:
\[ 6 \times 6 = 36 \equiv 1 \ (\text{mod} \ 7) \]

5. **Multiply Both Sides by the Inverse**:
Now multiply both sides of \( 6k \equiv 5 \) by 6:
\[ k \equiv 30 \ (\text{mod} \ 7) \]
Simplifying \( 30 \mod 7 \):
\[ 30 \div 7 = 4 \quad \text{remainder } 2 \]
So,
\[ k \equiv 2 \ (\text{mod} \ 7) \]

6. **Express \( k \)**:
We can express \( k \) as:
\[ k = 7m + 2 \quad \text{for some integer } m \]

7. **Substitute Back to Find \( x \)**:
Substitute \( k \) back into the equation for \( x \):
\[ x = 13(7m + 2) + 8 \]
\[ x = 91m + 26 + 8 \]
\[ x = 91m + 34 \]

8. **Find the Least Possible Value**:
To find the least possible value of \( x \), set \( m = 0 \):
\[ x = 34 \]

### Final Answer:
The least possible number is \( \boxed{34} \).