Solution:
(xⁿ - aⁿ) is divisible by (x - a) for all values of n
∴ (9⁶ - 1⁶) is divisible by (9 - 1)
⇒ (9⁶ - 1) is divisible by 8
⇒ on dividing (9⁶ + 1) by 8, we get 2 as remainder.

Solution:
$$\eqalign{ = \frac{{\left( {0.0539 - 0.002} \right) \times 0.4 + 0.56 \times 0.07}}{{0.04 \times 0.25}} \cr = \frac{{\left( {0.0519} \right) \times 0.4 + 0.56 \times 0.07}}{{0.04 \times 0.25}} \cr = \frac{{0.02076 + 0.0392}}{{0.04 \times 0.25}} \cr = \frac{{0.05996}}{{0.01}} \cr = 5.996 \cr} $$

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

Solution:
Let the numbers = a, b
According to the question,
a + b = 3 (a - b)
$$\frac{a + b}{a - b}$$   = $$\frac{3}{1}$$
By componendo and dividendo
$$\frac{2a}{2b}$$   = $$\frac{4}{2}$$
$$\frac{a}{b}$$   = $$\frac{2}{1}$$
a : b = 2 : 1

Solution:
Let numbers are 3x, 6x, 2x
Average : $$\frac{11x}{3}$$ = 44
x = 12 = Difference

Solution:
$$\eqalign{ {\left( {2153} \right)^{167}} \cr {\text{unit digit = }}{{\text{3}}^{167}} \cr {\text{unit digit }} \cr {{\text{3}}^1} \to {\text{3}} \to {\text{3}} \cr {{\text{3}}^2} \to 9 \to 9 \cr {{\text{3}}^3} \to 27 \to 7 \cr {{\text{3}}^4} \to 81 \to 1 \cr} $$
This cycle will continue
⇒ divide the power of 3 by 4
$$\eqalign{ \frac{{167}}{4} \Rightarrow {\text{remainder is 3}} \cr {{\text{3}}^3} \Rightarrow 7 \cr {\text{unit digit = }}1 \times 7 = {\text{ }}7 \cr} $$

Solution:
$$\eqalign{ = \left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{7}} \right) + \frac{1}{n} \cr = \frac{{\left( {21 + 14 + 6} \right)}}{{42}} + \frac{1}{n} \cr = \left( {\frac{1}{{42}} + \frac{1}{n}} \right) \cr} $$
This sum is a natural number when n = 42
So, each one of the statements that 2 divides ; 3 divides n and 7 divides n is true.
Hence, n > 84 is false