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This is one of our many quizzes on Real number. Click on this to access more such competitive quizzes on Number System.

Real number Quiz 2

mathematics

 A number of the form ${3^n}$, where $n$ is a natural number, cannot have the end digit as

 The last digit gets repeated after a few values of $n$.

${3^1} = 3, \\ {3^2} = 9, \\ {3^3} = 27, \\ {3^4} = 81, \\ {3^5} = 243… \\ $ The last digit will get repeated further.

 If $n$ is a natural number, then $\left( {{n^2} - n} \right)$ is always divisible by

 Product of an even and odd number is even.

$\left( {{n^2} - n} \right) = n\left( {n - 1} \right) \\ $ This is the product of two consecutive terms. Thus one of them will be an even number.

 The greatest 5-digit number divisible by $11$, $24$ and $35$ is

 The required number will be a multiple of the LCM of the numbers just less than $99999$.

LCM $\left( {11,24,35} \right) = 11 \times 24 \times 35 = 9240 \\ 99999 = \left( {9240 \times 10} \right) + 7599 \\ $ Thus, the required result is $9240 \times 10 = 92400$

 Flooring of a room requires square tiles to be fitted. The floor of the room measures $36ft \times 20ft$. What is the largest size of the tiles that can be used and how many such tiles will be required?

 The size of the tile has to exactly fit the length and breadth of the room.

The length of the square tile should fit both the length and breadth of the room without leaving any gap. $ \\ $ Thus, required size of the tile is HCF $\left( {36,20} \right) = 4ft$ $ \\ $ Number is such tiles required is $\dfrac{{36 \times 20}}{{4 \times 4}} = 9 \times 5 = 45$

 A number when divided by $24$ and $28$ leaves remainder as $7$ and $11$ respectively. The smallest such number is

 Write the LCM in the form of $LCM - x = pq + r$, where $p$ and $r$ are given and the result would be $\left( {LCM - x} \right)$.

$LCM\left( {24,28} \right) = 168 \\ $ Now, $168 = 161 + 7 \\ \Rightarrow 168 = \left( {24 \times 6 + 17} \right) + 7 \\ \Rightarrow 168 - 17 = \left( {24 \times 6} \right) + 7 \\ \Rightarrow 151 = \left( {24 \times 6} \right) + 7 \\ $ Similarly, $ 168 = 157 + 11 \\ \Rightarrow 168 = \left( {28 \times 5 + 17} \right) + 11 \\ \Rightarrow 168 - 17 = \left( {28 \times 5} \right) + 11 \\ \Rightarrow 151 = \left( {28 \times 5} \right) + 11 \\ $ Thus, $151 = \left( {24 \times 6} \right) + 7 = \left( {28 \times 5} \right) + 11$