(6n² + 6n) = 6n(n + 1), which is always divisible by 6 and 12 both, since n(n + 1) is always even.

107 x 107 + 93 x 93 = (107)² + (93)²

= (100 + 7)² + (100 - 7)²

= 2 x [(100)² + 7²]    [Ref: (a + b)² + (a - b)² = 2(a² + b²)]

= 20098

(xⁿ + 1) will be divisible by (x + 1) only when n is odd.

(67⁶⁷ + 1) will be divisible by (67 + 1)

(67⁶⁷ + 1) + 66, when divided by 68 will give 66 as remainder.

Largest 4 digit number = 9999
9999 ÷ 88 = 113, remainder = 55
Hence largest 4 digit number exactly divisible by 88 = 9999 - 55 = 9944

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Here, the given statement is like (a + b)² - 4ab where a= 481 and b = 426
(a + b)² - 4ab = (a² + 2ab + b²) - 4ab = a² - 2ab + b² = (a - b)²
Hence {(481 + 426)² - 4 x 481 x 426} = (481 - 426)² = 55² = 3025

To solve the expression {(481 + 426)^2 - 4 x 481 x 426} = ?, we need to simplify the expression step by step.

Let's start by simplifying the square inside the brackets:

• (481 + 426)^2 = 907^2

= 907 * 907

= 824649

Next, we need to simplify the expression inside the curly braces:

• {(481 + 426)^2 - 4 x 481 x 426} = 824649 - 4 x 481 x 426

= 824649 - 4 x (481 x 426)

= 824649 - 4 x 203866

= 824649 - 815544

= 3025

So the expression simplifies to 3025.

Therefore, the correct answer is A. 3025.

Let's summarize the solution in bullet points:

• Simplify the square inside the brackets: (481 + 426)^2 = 907^2 = 907 * 907 = 824649
• Simplify the expression inside the curly braces: {(481 + 426)^2 - 4 x 481 x 426} = 824649 - 4 x 481 x 426 = 824649 - 4 x (481 x 426) = 824649 - 4 x 203866 = 824649 - 815544 = 3025
• Hence, the expression simplifies to 3025, and the correct answer is A. 3025.

(a - b)² = a² - 2ab + b²

Here, the given statement is like (a - b)² + 4ab where a= 64 and b = 12
(a - b)² + 4ab = (a² - 2ab + b²) + 4ab = a² + 2ab + b² = (a + b)²
Hence (64 - 12)² + 4 x 64 x 12 = (64 + 12)² = 76² = 5776

Given Expression: (64 - 12)² + 4 × 64 × 12

Using BODMAS rule, we first perform the operation inside the bracket and then simplify the expression further.

64 - 12 = 52

So, the expression simplifies to:

(52)² + 4 × 64 × 12

= 2704 + 3072

= 5776

Hence, the correct answer is option D, 5776.

Let's understand the BODMAS rule and the concepts used in this problem in more detail:

• BODMAS Rule: BODMAS stands for Bracket, Order, Division, Multiplication, Addition, and Subtraction. It is a rule used to simplify mathematical expressions with multiple operations. According to the rule, we must perform the operations in the following order:
• Operations within the bracket
• Exponents or powers
• Division or multiplication (whichever comes first from left to right)
• Addition or subtraction (whichever comes first from left to right)
• Squaring: Squaring a number means multiplying it by itself. For example, the square of 4 is 4 × 4 = 16. We can also represent the square of a number using the exponent notation as a², where a is the number and ² is the exponent.
• Order of Operations: The order of operations refers to the order in which we perform mathematical operations to evaluate an expression. BODMAS is one such order of operations that helps us evaluate complex expressions in a systematic manner.
• Multiplication: Multiplication is an arithmetic operation that involves combining two or more numbers to get a product. We use the symbol "×" to denote multiplication.
• Addition: Addition is an arithmetic operation that involves combining two or more numbers to get a sum. We use the symbol "+" to denote addition.

By using the above concepts and following the BODMAS rule, we have simplified the given expression to get the answer as 5776.

Let 2³² = x. Then (2³² + 1) = (x + 1)
Assume that (x + 1) is completely divisible by a whole number, N
(2⁹⁶ + 1) = {(2³²)³ + 1} = (x³ + 1) = (x + 1)(x² - x + 1)
if (x + 1) is completely divisible by N, (x + 1)(x² - x + 1) will also be divisible by N
Hence (2⁹⁶ + 1) is completely divisible N

2 is even prime number

\(\frac{a^{2}+b^{2}-ab}{a^{3}+b^{3}}=\frac{(a^{2}+b^{2}-ab)}{(a+b)(a^{2}-ab+b^{2}}=\frac{1}{a+b}\)

\(\frac{1}{719+347} = \frac{1}{1066} \)

Let x be the smallest whole number in place of * 
Given that 481*673 to be completely divisible by 9, 
=> (4 + 8 + 1 + x + 6 + 7 + 3) is divisible by 9 (Reference : Divisibility by 9)
=> (29 + x) is divisible by 9
x should be the smallest whole number, Hence, (29 + x) = 36 
=> x = 36 - 29 = 7