Number System - Aptitude MCQ Questions and Solutions with Explanations

63592 is not a square of any natural number
∴ Any number that have 2, 3, 7, 8 on its units place , It could not be perfect square of any number.

$$\eqalign{ & \frac{1}{3}{\text{,}}\frac{4}{7},\frac{2}{5} \cr & \frac{1}{3} = 0.33,{\text{ }} \cr & \frac{4}{7} = 0.5, \cr & \frac{2}{5} = 0.4 \cr & \frac{1}{3} < \frac{2}{5} < \frac{4}{7} \cr} $$

$$\eqalign{ & = {17^{200}} \div 18 \cr & = {\left( {18 - 1} \right)^{200}} \div 18 \cr} $$
Apply Binomial theorem
$$ = {\left( {18} \right)^{200}}{\left( { - 1} \right)^0} + {\left( {18} \right)^{199}}{\left( { - 1} \right)^1} + $$       $$..... + \, {\left( {18} \right)^1}{\left( { - 1} \right)^{199}}$$   $$ \, + \, {\left( {18} \right)^0}{\left( { - 1} \right)^{200}}$$

$$ \Rightarrow $$ Remainder always comes from last term is (18)⁰ (- 1)²⁰⁰
$$\eqalign{ & = \frac{{{{\left( {18} \right)}^0}{{\left( { - 1} \right)}^{200}}}}{{18}} \cr & = 1 \times {1^{200}} \cr & = 1 \cr & {\text{So, remainder}} = 1 \cr} $$

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

$$\eqalign{ & {\text{Let the number is }}x{\text{}} \cr & {\text{According to question,}} \cr & {\text{3}}\left( {2x + 9} \right) = 75 \cr & 2x + 9 = 25 \cr & x = \frac{{16}}{2} \cr & x = 8 \cr} $$

$$\eqalign{ & {\text{Let number of boys}} = x \cr & {\text{Number of 25 paise coins}} = {x^2} \cr & {\text{According to question,}} \cr & \frac{{{x^2}}}{4} = 400 \cr & {x^2} = 1600 \cr & x = 40\,\, \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & {{\text{n}}^4} + 6{{\text{n}}^3} + 11{{\text{n}}^2} + 6{\text{n + 24}} \cr & {\text{put n = 1}} \cr & = 1 + 6 + 11 + 6 + 24 \cr & = 48 \cr & {\text{put n = 2}} \cr & {\text{ = 16 + 48 + 44 + 12 + 24}} \cr & {\text{ = 144}} \cr & {\text{Clearly it is divisible by 48}} \cr} $$

The sum of all natural numbers upto 97 - sum of all natural numbers upto 74
sum of n natural numbers
$$\eqalign{ & = \frac{{n\left( {n + 1} \right)}}{2}\,\,\left( {{\text{Formula}}} \right) \cr & = \frac{{97 \times 98}}{2} - \frac{{74 \times 75}}{2} \cr & = 4753 - 2775 \cr & = 1978 \cr} $$

$$\eqalign{ & \frac{4}{5}{\text{ of an estate = 16800}} \cr & {\text{1 of an estate = 16800}} \times \frac{5}{4} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21000 \cr & \frac{3}{7}{\text{ of an estate = 21000}} \times \frac{3}{7} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 9000 \cr} $$

A number will divisible by 18 if it is divisible by 2 and 9
Clearly we can see 65043 is not divisible by 2
Because unit digit of 65043 is 3 so this will not be divisible by 18