Basic Maths: Test 38

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Basic Maths: Test 38

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Question 1

Which one of the following does not divide$\left( {{5}^{2430}}-1 \right)$?

A	24
B	124
C	624
D	3124

Question 1 Explanation:

When n is odd or even, ${{x}^{n}}-{{a}^{n}}$ is exactly divisible by x-a. $\begin{align} & \left( {{5}^{2430}}-1={{25}^{1215}}-1,\,\,25-1\,\,\,i.e.\,\,is\,\,\,divisible\,\,by\,\,24 \right) \\ & {{5}^{2430}}-1={{\left( {{5}^{3}} \right)}^{810}}-1={{125}^{810}}-1, \\ & which\,\,\,\,is\,\,\,\,divisible\,\,by\,\,124. \\ & {{5}^{2430}}-1={{\left( {{5}^{5}} \right)}^{486}}-1, \\ & ={{\left( 3125 \right)}^{486}}-1\,\,which\,\,\,is\,\,\,divisible\,\,\,\,by\,3124. \\ \end{align}$

Question 2

When 10³¹ is divided by 999, the remainder is

A	1
B	10
C	100
D	999

Question 2 Explanation:

Question 3

Simplify: ${{\left( 0\overline{.2} \right)}^{2}}\left\{ \,3-11{{\left( 0.\overline{27} \right)}^{2}} \right\}$

A	$-\frac{32}{162}$
B	$\frac{1}{297}$
C	$\frac{76}{99}$
D	$\frac{32}{297}$

Question 3 Explanation:

$\begin{align} & {{\left( \frac{2}{9} \right)}^{2}}\left\{ 3-11{{\left( \frac{27}{99} \right)}^{2}} \right\} \\ & =\frac{4}{81}\left\{ 3-\frac{11\times 3\times 3}{11\times 11} \right\} \\ & =\frac{4}{81}\times \left\{ 3-\frac{9}{11} \right\} \\ & =\frac{4}{81}\times \left\{ \frac{24}{11} \right\} \\ & =\frac{4}{81}\times \frac{24}{11}=\frac{32}{297} \\ \end{align}$

Question 4

${{\left( 28 \right)}^{3}}+{{\left( 75 \right)}^{3}}-{{\left( 103 \right)}^{3}}+3\times 28\times 75\times 103$ is equal to:

A	10000
B	0
C	30007
D	1

Question 4 Explanation:

$\begin{align} & Let,\,\,28=\,\,a \\ & 75=\,b,\,-103=c \\ & When\,\,\left( a+b+c \right)=0 \\ & {{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=0 \\ & Here,\,\,a+b+c=\,28+75-103=0 \\ & Therefore\,\,{{(28)}^{3}}+{{\left( 75 \right)}^{3}}-{{\left( 103 \right)}^{3}}+3\times 28\times 75\times 103=0 \\ \end{align}$

Question 5