What is the value of `1006^(2) - 1007 xx 1005 + 1008 xx 1004 - 1009 xx 1003` ?

Options

A. 6
B. 3
C. 12
D. 24

Answer

To solve the expression \( 1006^2 - 1007 \times 1005 + 1008 \times 1004 - 1009 \times 1003 \), we can simplify it step by step.

### Step 1: Calculate \( 1006^2 \)
\[ 1006^2 = 1006 \times 1006 = 1012036 \]

### Step 2: Calculate \( 1007 \times 1005 \)
\[ 1007 \times 1005 = 1007 \times (1000 + 5) = 1007 \times 1000 + 1007 \times 5 = 1007000 + 5035 = 1002035 \]

### Step 3: Calculate \( 1008 \times 1004 \)
\[ 1008 \times 1004 = 1008 \times (1000 + 4) = 1008 \times 1000 + 1008 \times 4 = 1008000 + 4032 = 1002032 \]

### Step 4: Calculate \( 1009 \times 1003 \)
\[ 1009 \times 1003 = 1009 \times (1000 + 3) = 1009 \times 1000 + 1009 \times 3 = 1009000 + 3027 = 1006027 \]

### Step 5: Substitute the values into the expression
Now we substitute all the calculated values back into the original expression:
\[ 1006^2 - 1007 \times 1005 + 1008 \times 1004 - 1009 \times 1003 = 1012036 - 1002035 + 1002032 - 1006027 \]

### Step 6: Simplify the expression
Now, we simplify step by step:
1. Calculate \( 1012036 - 1002035 \):
\[ 1012036 - 1002035 = 1001 \]
2. Now add \( 1002032 \):
\[ 1001 + 1002032 = 1004033 \]
3. Finally, subtract \( 1006027 \):
\[ 1004033 - 1006027 = -1994 \]

### Final Answer
Thus, the value of the expression \( 1006^2 - 1007 \times 1005 + 1008 \times 1004 - 1009 \times 1003 \) is:
\[ \boxed{-1994} \]