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The number 111 \ldots \ldots .1  (91 times) is
 

Option: 1

Not a prime 


Option: 2

An even number


Option: 3

Not an odd number


Option: 4

None of these


Answers (1)

best_answer

111 \ldots . .1 (91\ times)\\\\ \begin{aligned} & =1+10+10^2+\ldots .+10^{90} \\ & =\frac{10^{91}-1}{10-1}=\frac{\left(10^7\right)^{13}-1}{10-1}=\frac{t^{13}-1}{9}, \text { where } t=10^7 \\ & =\left(\frac{t-1}{9}\right)\left(t^{12}+t^{11}+\ldots . .+t+1\right) \\ & =\left(\frac{10^7-1}{10-1}\right)\left(1+t+t^2+\ldots .+t^{12}\right) \\ & =\left(1+10+10^2+\ldots .+10^6\right)\left(1+t+t^2+\ldots+t^{12}\right) \end{aligned}\\\\ \therefore 111 \ldots . .1 (91\ times)\ \text{is a composite number.}

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Ritika Kankaria

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