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

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

Clearly, π is an irrational number

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

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

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.

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

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

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.

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

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

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

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

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

All natural numbers and 0 are called the whole numbers.

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

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

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

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

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