When `{:((D)/(N)toR=14, "option " 5),((3D)/(N)toR=8," "6),((4D)/(N)toR=?," "8):}`

Options

A. 5
B. 6
C. 8
D. 4

Answer

To solve the problem step by step, we need to analyze the information given about the number \( D \), the divisor \( N \), and the remainders \( R \).

### Step 1: Understand the given information
We have:
1. \( \frac{D}{N} \) gives a remainder \( R = 14 \)
2. \( \frac{3D}{N} \) gives a remainder \( R = 8 \)
3. We need to find the remainder when \( \frac{4D}{N} \).

### Step 2: Express the relationships mathematically
From the first condition, we can express \( D \) in terms of \( N \) and \( R \):
\[ D = kN + 14 \]
for some integer \( k \).

From the second condition:
\[ 3D = mN + 8 \]
for some integer \( m \).

### Step 3: Substitute \( D \) into the second equation
Substituting \( D \) from the first equation into the second:
\[ 3(kN + 14) = mN + 8 \]
Expanding this gives:
\[ 3kN + 42 = mN + 8 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 3kN - mN = 8 - 42 \]
\[ (3k - m)N = -34 \]

### Step 5: Analyze possible values for \( N \)
Since \( N \) must be a divisor of \( -34 \), the possible positive divisors of \( 34 \) are \( 1, 2, 17, 34 \).

### Step 6: Check each divisor
1. **If \( N = 1 \)**: \( D \) would yield a remainder of \( 0 \) which is not possible.
2. **If \( N = 2 \)**: This would not satisfy the conditions as the remainders would not match.
3. **If \( N = 17 \)**:
- For \( D = k \cdot 17 + 14 \)
- \( 3D = 3(k \cdot 17 + 14) = 51k + 42 \)
- The remainder when dividing \( 51k + 42 \) by \( 17 \) is \( 8 \) (which is correct).
4. **If \( N = 34 \)**:
- The remainder calculations would not yield valid results.

### Step 7: Calculate the remainder for \( 4D \)
Now we calculate \( 4D \):
\[ D = k \cdot 17 + 14 \]
Thus,
\[ 4D = 4(k \cdot 17 + 14) = 68k + 56 \]

### Step 8: Find the remainder when dividing \( 4D \) by \( N \)
Now, we need to find \( 4D \mod 17 \):
\[ 68k + 56 \mod 17 \]
Since \( 68 \mod 17 = 0 \) (as \( 68 \) is a multiple of \( 17 \)):
\[ 4D \mod 17 = 56 \mod 17 \]
Calculating \( 56 \mod 17 \):
- \( 56 \div 17 = 3 \) remainder \( 5 \).

Thus, the remainder when \( 4D \) is divided by \( N \) is \( 5 \).

### Final Answer
The remainder when \( \frac{4D}{N} \) is \( 5 \).

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