Let d + 2(2b + c)=19. What will be the remainder when the 4-digit number abcd is divided by 8?

Options

A. 2
B. 5
C. 0
D. 3

Answer

To solve the problem, we need to find the remainder when the 4-digit number \( abcd \) is divided by 8, given the equation \( d + 2(2b + c) = 19 \).

### Step-by-Step Solution:

1. **Understanding the 4-digit number**:
The 4-digit number \( abcd \) can be expressed as:
\[ abcd = 1000a + 100b + 10c + d \]

2. **Divisibility by 8**:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. Therefore, we only need to consider the last three digits \( bcd \):
\[ bcd = 100b + 10c + d \]

3. **Using the given equation**:
We have the equation:
\[ d + 2(2b + c) = 19 \]
This can be rewritten as:
\[ d + 4b + 2c = 19 \]
Rearranging gives:
\[ d = 19 - 4b - 2c \]

4. **Substituting \( d \)**:
We substitute \( d \) back into the expression for \( bcd \):
\[ bcd = 100b + 10c + (19 - 4b - 2c) \]
Simplifying this:
\[ bcd = 100b + 10c + 19 - 4b - 2c = (100b - 4b) + (10c - 2c) + 19 \]
\[ bcd = 96b + 8c + 19 \]

5. **Finding the remainder when divided by 8**:
Now, we need to find the remainder of \( bcd \) when divided by 8:
\[ bcd \mod 8 = (96b + 8c + 19) \mod 8 \]
Since \( 96b \) and \( 8c \) are both multiples of 8, they will give a remainder of 0 when divided by 8:
\[ bcd \mod 8 = 0 + 0 + 19 \mod 8 \]
Now, we compute \( 19 \mod 8 \):
\[ 19 \div 8 = 2 \quad \text{(remainder 3)} \]
Thus:
\[ 19 \mod 8 = 3 \]

6. **Final Answer**:
The remainder when the 4-digit number \( abcd \) is divided by 8 is:
\[ \boxed{3} \]