Question

Determine whether the number 1111...1 of 91 digits is prime or composite ?

Answer 1

Let $N=\mathrm{111111.......}=1+10+{10}^2+{10}^3+{10}^4+.............{.10}^{90}=\frac{{10}^{91}-1}{9}$
Now ${10}^{91}=(9+1{)}^{91}{=}^{91}{C}_0{9}^{91}{+}^{91}{C}_1{9}^{90}+.............{+}^{91}{C}_{1}9+1$
Thus $N{=}^{91}{C}_0{9}^{90}{+}^{91}{C}_1{9}^{89}+..................{+}^{91}{C}_{2}9{+}^{91}{C}_1$
Hence given number is composite.