7.Binomial Theorem
hard

The number $111......1 $ ( $ 91$ times) is

A

Not a prime

B

An even number

C

Not an odd number

D

None of these

Solution

(a) $111…..1$ ($91$ times)

= $1 + 10 + {10^2} + ….. + {10^{90}}$

= $\frac{{{{10}^{91}} – 1}}{{10 – 1}} = \frac{{{{({{10}^7})}^{13}} – 1}}{{10 – 1}}$= $\frac{{{t^{13}} – 1}}{9}$, where $t = {10^7}$

= $\left( {\frac{{t – 1}}{9}} \right)\,({t^{12}} + {t^{11}} + ….. + t + 1)$

= $\left( {\frac{{{{10}^7} – 1}}{{10 – 1}}} \right)\,(1 + t + {t^2} + …. + {t^{12}})$

$ = (1 + 10 + {10^2} + …. + {10^6})(1 + t + {t^2} + … + {t^{12}})$

 $111…..1(91\,\,{\rm{times)}}$ is a composite number.

Standard 11
Mathematics

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