Since 13 is factor of 52. So divide its remainder directly by 13.
$$\eqalign{ & \Rightarrow \frac{{{\text{remainder}}}}{{13}} = \frac{{45}}{{13}} \cr & \Rightarrow {\text{remainder }} = 6 \cr} $$

given $${\text{ }}\frac{3}{4}$$ = 0.75     $${\text{ }}\frac{3}{8}$$ = 0.375
(a) $${\text{ }}\frac{7}{9}$$ → 0.77
(b) $${\text{ }}\frac{7}{3}$$ → 2.3
(c) $${\text{ }}\frac{5}{9}$$ → 0.55
(d) $${1\text{ }}\frac{9}{16}$$ = $${\text{ }}\frac{25}{16}$$ → 1.5
$${\text{ }}\frac{5}{9}$$ = 0.55

Let the numerator and denominator x and y
According to question,
$$\eqalign{ & \frac{{x + 1}}{{y + 1}} = \frac{1}{4} \cr & 4x + 4 = y + 1 \cr & 4x - y = - 3.....(i) \cr & \frac{{x + 2}}{{y + 2}} = \frac{1}{3} \cr & 3x + 6 = y + 2 \cr & 3x - y = - 4.....(ii) \cr & {\text{Solve (i) and (ii) }} \cr & x = 1\,\,\,\& \,\,\,y = 7 \cr} $$
Sum of numerator and denominator of fraction :
= x + y
= 1 + 7
= 8

First multiple = 3
Last multiple (less than 50) = 48
Number of multiples of 3
= $$\frac{48 - 3}{3}$$ + 1
= 16
Sum :
= $$\frac{16}{2}$$ [2 × 3 + (16 - 1)3]
= 8 × 51
= 408

Least number of 5 digit is 10000
10000 is divided by 123
then the remainder is 37
Least number that divisible by 123
=10000 − 37 + 123
=10086

In this case:
122 (powers of the last digit 2 )

21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
……

As you can see, the pattern repeats every 4th power (2, 4, 8, 6).

We have to find 122173
Hence, remainder when 173 is divided by 4 = 1
(which corresponds to the 1st number of the pattern)
Thus, unit digit of 122173 = 2

Let the number be x
According to question
$$\eqalign{ & \left( {3 \times {x^2}} \right){\text{ - }}\left( {4 \times x} \right) = 50 + x.....(i) \cr & \Rightarrow 3{x^2} - 4x = 50 + x \cr & \Rightarrow 3{x^2} - 5x - 50 = 0 \cr & \Rightarrow 3{x^2} - 15x + 10x - 50 = 0 \cr & \Rightarrow 3{x^{}}(x - 5) + 10(x - 5) = 0 \cr & \Rightarrow (x - 5)(3x + 10) = 0 \cr & x = 5\,\,or\,\, - \frac{{10}}{3} \cr} $$
Since the natural number is x = 5

Shortcut method :
$$ \Rightarrow 3{x^2} - 4x = 50 + x.....(i)$$
Now put the value of x from option (b)
$$\eqalign{ & x = 5 \cr & 3 \times {(5)^2} - 4 \times 5 = 50 + 5 \cr & 75 - 20 = 55 \cr & 55 = 55 \cr} $$
LHS = RHS (it satisfies the conditions)
$${\text{so, x = 5}}$$

$$\eqalign{ & \Rightarrow {2^{16}} - 1\,\, \cr & = \left( {{2^8} - {1^8}\,\,} \right)\left( {{2^8} + {1^8}\,} \right) \cr & = \left( {{2^4} - 1\,\,} \right)\left( {{2^4} + 1\,\,} \right)\left( {{2^8} + 1\,\,} \right) \cr & = \left( {16 - 1} \right)\left( {16 + 1} \right)\left( {{2^8} + 1\,\,} \right) \cr & = 15 \times 17\left( {{2^8} + 1\,\,} \right) \cr & \therefore {2^{16}} - 1\,\,{\text{is divisible by 17}} \cr} $$

Let the ten's digits be x
Unit digit be 2x - 1
Original number :
= 10x + 2x - 1
= 12x - 1
New number :
= 10(2x - 1) + x
= 20x - 10 + x
= 21x - 10
According to the question,
(21x - 10) - (12x - 1) = (12x - 1) - 20
9x - 9 = 12x - 21
3x = 12
x = 4
Original number :
= 12x - 1
= 12 × 4 - 1
= 47

$$999\,\frac{1}{7} + 999\frac{2}{7} + 999\frac{3}{7} + $$     $$999\frac{4}{7} + $$   $$999\frac{5}{7} + $$   $$999\frac{6}{7}$$
$$ \Rightarrow \left( {999 \times 6} \right) + $$   $$\left( {\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \frac{4}{7} + \frac{5}{7} + \frac{6}{7}} \right)$$
$$\eqalign{ & \Rightarrow 5994 + \left( {\frac{{21}}{7}} \right) \cr & \Rightarrow 5994 + 3 \cr & \Rightarrow 5997 \cr} $$

Unit's digit in 3 x 38 x 537 x 1256
= Unit's digit in 3 x 8 x 7 x 6
= 4 x 2
= 8
Note : Always multiply only unit digit of first number, to second and product's unit digits number with 3rd number. Again product of last's unit digit to fourth and so on.

$$\eqalign{ & \Rightarrow \left( {0.\overline {63} + 0.\overline {37} } \right) \cr & = \frac{{63}}{{99}} + \frac{{37}}{{99}} \cr & = \frac{{63 + 37}}{{99}} \cr & = \frac{{100}}{{99}} \cr} $$

Man eats G grapes on 1st day
According to question
G + (G + 6) + (G + 12) + (G + 18) + (G + 24) = 100
5G + 60 = 100
5G = 40
G = 8

Square root
$$ \Rightarrow \sqrt {\frac{{\left( {0.064 - 0.008} \right)\left( {0.16 - 0.04} \right)}}{{\left( {0.016 + 0.08 + 0.04} \right){{\left( {0.4 + 0.2} \right)}^3}}}} $$
$$ \Rightarrow \sqrt {\frac{{\left[ {{{\left( {0.4} \right)}^3} - {{\left( {0.2} \right)}^3}} \right]\left[ {{{\left( {0.4} \right)}^2} - {{\left( {0.2} \right)}^2}} \right]}}{{\left[ {{{\left( {0.4} \right)}^2} + {{\left( {0.2} \right)}^2} + \left( {0.2 \times 0.4} \right)} \right]{{\left[ {\left( {0.4} \right) + 0.2} \right]}^3}}}} $$
$$ \Rightarrow {\text{Let }}0.4{\text{ }} = {\text{ }}a,{\text{ }}0.2{\text{ }} = {\text{ }}b$$
$$ \Rightarrow \sqrt {\frac{{\left( {{a^3} - {b^3}} \right)\left( {{a^2} - {b^2}} \right)}}{{\left( {{a^2} + {b^2} + ab} \right){{\left( {a + b} \right)}^3}}}} $$
$$ \Rightarrow \sqrt {\frac{{\left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)\left( {a + b} \right)\left( {a - b} \right)}}{{{{\left( {a + b} \right)}^3}\left( {{a^2} + {b^2} + ab} \right)}}} $$
$$\eqalign{ & \Rightarrow \sqrt {\frac{{{{\left( {a - b} \right)}^2}}}{{{{\left( {a + b} \right)}^2}}}} \cr & \Rightarrow \frac{{a - b}}{{a + b}} \cr & \Rightarrow \frac{{0.4 - 0.2}}{{0.4 + 0.2}} \cr & \Rightarrow \frac{{0.2}}{{0.6}} \cr & \Rightarrow \frac{1}{3} \cr} $$

$$\eqalign{ & = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{{14}} + \frac{1}{{28}} \cr & = \frac{{28 + 14 + 7 + 4 + 2 + 1}}{{28}} \cr & = \frac{{56}}{{28}} \cr & = 2 \cr} $$

5 * 7
= 5 + 7 - 5 × 7
= 12 - 35
= - 23

a = 4011
b = 3989
a × b = 4011 × 3989
         = 15999879

Shortcut method :
a = 4011 ⇒ (4000 + 11)
b = 3989 ⇒ (4000 - 11)
[(a - b) (a + b) = a2 - b2]
According to the question,
a × b = (4000 + 11) (4000 - 11)
        = 40002 -112
        = 16000000 - 121
        = 15999879

Sum = 5 + 10 + 15 + 20 + .....+ 95 , It is an arithmetic progression
Number of terms (n)
= $$\frac{{{\text{Last term }} - {\text{ First term}}}}{{{\text{Differnce}}}}$$     + 1
⇒ n = $$\frac{95 - 5}{5}$$ + 1 = 19
So, sum = $$\frac{n}{2}$$ [2a + (n - 1)d]
Here, a = 5, n = 19, d = 5
Sum = $$\frac{19}{2}$$ [10 + (19 - 1)5]
        = 950

5349 + 3957 - 7062 = 2244
This is not divisible by 7.

We know,
Smallest five digit number is = 10000
L.C.M. of 12, 18, 21 = 252
10000 is divided by 252
then the remainder is 172
Difference = 252 - 172 = 80
Then number is = 10000 + 80 = 10080