If positive numbers x, y and z are divided by 28. The remainders are 13, 19 and 23, respectively. Find the remainder when `(3x - 2y + 2z)` is divided by 14?

Options

A. 5
B. 7
C. 6
D. 8

Answer

To solve the problem, we need to find the remainder when \(3x - 2y + 2z\) is divided by 14, given the remainders of \(x\), \(y\), and \(z\) when divided by 28.

### Step 1: Express \(x\), \(y\), and \(z\) in terms of their remainders
Since \(x\), \(y\), and \(z\) are given as:
- \(x \equiv 13 \mod 28\)
- \(y \equiv 19 \mod 28\)
- \(z \equiv 23 \mod 28\)

We can express them as:
- \(x = 28q_1 + 13\)
- \(y = 28q_2 + 19\)
- \(z = 28q_3 + 23\)

where \(q_1\), \(q_2\), and \(q_3\) are the respective quotients when \(x\), \(y\), and \(z\) are divided by 28.

### Step 2: Substitute \(x\), \(y\), and \(z\) into the expression \(3x - 2y + 2z\)
Now we substitute these expressions into \(3x - 2y + 2z\):
\[ 3x - 2y + 2z = 3(28q_1 + 13) - 2(28q_2 + 19) + 2(28q_3 + 23) \]

### Step 3: Simplify the expression
Expanding this gives:
\[ = 3 \cdot 28q_1 + 39 - 2 \cdot 28q_2 - 38 + 2 \cdot 28q_3 + 46 \]
Combining the constant terms:
\[ = 3 \cdot 28q_1 - 2 \cdot 28q_2 + 2 \cdot 28q_3 + (39 - 38 + 46) \]
\[ = 3 \cdot 28q_1 - 2 \cdot 28q_2 + 2 \cdot 28q_3 + 47 \]

### Step 4: Factor out the common term
Now we can factor out 28:
\[ = 28(3q_1 - 2q_2 + 2q_3) + 47 \]

### Step 5: Find the remainder when divided by 14
Now we need to find the remainder of this expression when divided by 14:
\[ 28(3q_1 - 2q_2 + 2q_3) \mod 14 = 0 \quad (\text{since } 28 \text{ is a multiple of } 14) \]
Thus, we only need to consider the remainder of 47 when divided by 14:
\[ 47 \div 14 = 3 \quad \text{(which gives a quotient of 3)} \]
Calculating \(14 \times 3 = 42\), we find the remainder:
\[ 47 - 42 = 5 \]

### Final Answer
The remainder when \(3x - 2y + 2z\) is divided by 14 is \(5\).