Let the ten's digit be x and unit's digit be y.

Then, number = 10x + y.

Number obtained by interchanging the digits = 10y + x.

 therefore  (10x + y) + (10y + x) = 11(x + y), which is divisible by 11.

Clearly, 97 is a prime number.

Unit digit in (6374)¹⁷⁹³ = Unit digit in (4)¹⁷⁹³

    = Unit digit in [(4²)⁸⁹⁶ x 4]

    = Unit digit in (6 x 4) = 4

Unit digit in (625)³¹⁷ = Unit digit in (5)³¹⁷ = 5

Unit digit in (341)⁴⁹¹ = Unit digit in (1)⁴⁹¹ = 1

Required digit = Unit digit in (4 x 5 x 1) = 0.

5358 x 51 = 5358 x (50 + 1)

= 5358 x 50 + 5358 x 1

= 267900 + 5358

= 273258.

Required sum = (2 + 3 + 5 + 7 + 11) = 28.

Note: 1 is not a prime number.

Definition: A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.

Let the smaller number be x. Then larger number = (x + 1365).

Therefore  x + 1365 = 6x + 15

 5x = 1350

 x = 270

Threrfore Smaller number = 270.

Given Exp. =      \(\frac{\left(12\right)^{3}\times6^{4}}{432} =\frac{\left(12\right)^{3}\times6^{4}}{12\times6^{2}}\) = (12)² x 6² = (72)² = 5184

Solution:

To solve this question, we will use the basic rules of exponents and division.

•Exponents: In mathematics, an exponent is a number or symbol that is used to indicate the number of times a number is to be multiplied by itself. The exponent is written as a superscript number to the right of the base number. For example, if we have the expression 4^3, the base number is 4 and the exponent is 3. This expression is read as 4 to the power of 3, which is equal to 4 x 4 x 4 = 64.

•Division: Division is a mathematical operation which is the inverse of multiplication. It is the process of breaking apart a quantity into a number of equal parts. For example, when we divide 8 by 2, we get 4. This can be written as 8÷2 = 4.

Now let's solve the given problem:

(12)^3 x 6^4 ÷ 432 = ?

We will solve this problem by first simplifying the expression.

•Simplification:

(12)^3 x 6^4 ÷ 432 = (12x12x12) x (6x6x6x6) ÷ (4x4x4x4x4)

= 1728 x 1296 ÷ 1024

= 2239488 ÷ 1024

= 2184

Therefore, the answer is A. 5184.

72519 x 9999 = 72519 x (10000 - 1)

= 72519 x 10000 - 72519 x 1

= 725117481

To solve this question, we can use the multiplication formula for two numbers that differ by one, i.e., (a)(a+1) = a^2 + a. We can break down 9999 into 10000 - 1 and use this formula to simplify the multiplication.

72519 x (10000 - 1)

= 72519 x 10000 - 72519 x 1 (distributive property of multiplication)

= 725190000 - 72519 (simplifying)

= 725117481

Therefore, the correct answer is option A, 725117481.

Here are some relevant definitions and formulas that we used:

• Multiplication: a mathematical operation used for finding the total when a specific number is added to itself a certain number of times.
• Distributive property of multiplication: a(b + c) = ab + ac. This property allows us to multiply a sum by a number and get the same result as multiplying each addend by the number and then adding the products.
• Square: the result of multiplying a number by itself. For example, 3^2 = 3 x 3 = 9.
• Formula for multiplying two numbers that differ by one: (a)(a+1) = a^2 + a. This formula can be useful for simplifying multiplication when one of the numbers is close to a multiple of 10 or 100.

In this problem, we used the distributive property of multiplication and the formula for multiplying two numbers that differ by one to simplify the multiplication of 72519 and 9999.

Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.

Therefore x = 2.

The smallest 3-digit number is 100, which is divisible by 2.

 100 is not a prime number.

101 < 11 and 101 is not divisible by any of the prime numbers 2, 3, 5, 7, 11.

101 is a prime number.

Hence 101 is the smallest 3-digit prime number.