Question

Which of the following numbers is/are composite

• A. $\mathrm{1111.........1}\left(91\text{digits}\right)$
• B. $\mathrm{1111.........1}\left(61\text{digits}\right)$
• C. $\mathrm{1111.........1}\left(175\text{digits}\right)$
• D. $\mathrm{1111.........1}\left(105\text{digits}\right)$

Answer 1

The correct option is A $\mathrm{1111.........1}\left(91\text{digits}\right)$

Only $\mathrm{1111.......1}\left(61\text{digits}\right)$ is a prime number, all the remaining are composite.

For example,

$\mathrm{1111.......1}\left(91\text{digits}\right)$

$=\frac{1}{9}({10}^{91}-1)$

$=\frac{1}{9}({10}^7-1)\left(\right({10}^7{)}^{12}+({10}^7{)}^{11}+.....+1)$

$=\left(\mathrm{111.....1}\right(7\text{digits}\left)\right)\left(\right({10}^7{)}^{12}+({10}^7{)}^{11}+.....+1)$

Which is clearly a composite number.

Similarly, option (C) and (D) can be proved to be composite.