Every real number is a rational number is not a correct statements.

Clearly, π is an irrational number

We know that 11 and 13 are co-prime
So, a number divisible by both 11 and 13 will be divisible by (11 × 13)

On dividing 1056 by 23, we get 21 as remainder.
∴ Required number to be added = (23 - 21) = 2

$$\eqalign{ & \left( {287 \times 287 + 269 \times 269 - 2 \times 287 \times 269} \right) \cr & = {\left( {287} \right)^2} + {\left( {269} \right)^2} - \left( {2 \times 287 \times 269} \right) \cr & = {\left( {287 - 269} \right)^2} \cr & = {\left( {18} \right)^2} \cr & = 324 \cr} $$
[∵ a2 + b2 - 2ab = (a - b)2 ]

All natural numbers and 0 are called the whole numbers.

Let N = 5 × 10 × 15 × 20 × 25 × 30 × 35 × 40 × 45 × 50
= 510 × ( 1 × 2 × 3 × 4 × ..... × 10) = 510 × 10!
Highest power of 2 in 10!
$$ = \left[ {\frac{{10}}{2}} \right] + \left[ {\frac{{10}}{{{2^2}}}} \right] + \left[ {\frac{{10}}{{{2^3}}}} \right]$$
= 5 + 2 + 1
= 8
Highest power of 5 in 10! = $$\left[ {\frac{{10}}{5}} \right]$$ = 2
∴ N = 28 × 512 × k
Since highest power of 2 is less than that of 5,
So, required number of zeros = 8

The required arithmetic sequence of five prime numbers is 5, 11, 17, 23, 29 and therefore, the required 5th term is 29

Let,
8888 + 848 +88 - x = 7337 + 737
Then,
9824 - x = 8074
x = 9824 - 8074
x = 1750

Required sum
$$\eqalign{ & = \frac{{100}}{2}\left[ {1 + 100} \right] \cr & = 101 \times 50 \cr & = 5050 \cr} $$

Let, 5566 - 7788 + 9988 = x + 4444
Then,
(5566 + 9988) - 7788 = x + 4444
⇒ 15554 - 7788 = x + 4444
⇒ x + 4444 = 7766
⇒ x = (7766 - 4444)
⇒ x = 3322

Let the required number be x.
Then, $$\frac{11x + 11}{13}$$   = a whole number.
So, (11x + 11) must be divisible by 13
By hit and trial, we get x = 12
Hence, the smallest original number is 12

(+ 1) - (- 1)
= (+ 1 + 1)
= 2

Since (a + b)2 - (a - b)2 = 4ab, So, the given expression should be a multiple of 4
So, the least value of 4ab is 32 and so the least value of ab is 8
Hence, the smallest value of a is 4 and that of b is 2
Hence, a = 4

z = 3 × 31 + 2 = 95
y = 7 × z + 4 = 7 × 95 + 4 = 669
x = 8 × y + 3 = 8 × 669 + 3 = 5355
∴ Required number = 5355

It is given that x - y = 8
I. We may have x = 5 and y = - 3
So, it is not necessary that both x and y are positive.
II. If x positive, then it is not necessary that y is positive, as x = 5 and y = -3
III. If x < 0, then y = x - 8 which is clearly less than 0
So, if x is negative, then y must be negative.

Since, $$\sqrt 2 $$  is a non-terminating and non-repeating decimal, so it is an irrational number.

9H + H8 + H6 = 230
⇒ {(9 × 10) + H} + (10H + 8) + (10H + 6) = 230
⇒ 21H + 104 = 230
⇒ 21H = 126
⇒ H = 6

When n is even, (xn - an) is divisible by (x + a)
Now, 2256 = (24)64 = (16)64
∴ (1664 - 164) is divisible by (16 + 1)
⇒ (1664 - 1) is divisible by 17
⇒ (2256 - 1) is divisible by 17
⇒ On dividing 2256 by 17, we get 1 as remainder.

Let the hundreds, tens and unit digits be x, y and z respectively.
Then,
(100x + 10y + z) - (x + y + z)
= 99x + 9y
= 9 (11x + y)
So, the resulting number is divisible by 9