What is the value of N if `N=624^(2)+625xx623-628xx620-630xx618`

Options

A. 50
B. 51
C. 54
D. 55

Answer

To solve for \( N \) in the equation \( N = 624^2 + 625 \times 623 - 628 \times 620 - 630 \times 618 \), we can simplify the expression step by step.

### Step 1: Expand the Expression
Start with the original expression:
\[ N = 624^2 + 625 \times 623 - 628 \times 620 - 630 \times 618 \]

### Step 2: Rewrite the Products
Notice that \( 625 = 624 + 1 \), \( 623 = 624 - 1 \), \( 628 = 624 + 4 \), \( 620 = 624 - 4 \), \( 630 = 624 + 6 \), and \( 618 = 624 - 6 \). We can rewrite the products:
\[ 625 \times 623 = (624 + 1)(624 - 1) = 624^2 - 1 \]
\[ 628 \times 620 = (624 + 4)(624 - 4) = 624^2 - 16 \]
\[ 630 \times 618 = (624 + 6)(624 - 6) = 624^2 - 36 \]

### Step 3: Substitute Back into the Expression
Now substitute these back into the expression for \( N \):
\[ N = 624^2 + (624^2 - 1) - (624^2 - 16) - (624^2 - 36) \]

### Step 4: Simplify the Expression
Now simplify:
\[ N = 624^2 + 624^2 - 1 - 624^2 + 16 - 624^2 + 36 \]
Combine like terms:
\[ N = 624^2 + 624^2 - 624^2 - 624^2 - 1 + 16 + 36 \]
This simplifies to:
\[ N = -1 + 16 + 36 \]
\[ N = 51 \]

### Final Answer
Thus, the value of \( N \) is:
\[ \boxed{51} \]