<s>[INST] Prove that number made only of $91$ ones is composite How to prove that number  $11...1$ (there are exactly $91$ ones) is composite? I considered this number modulo some small primes, but none of them worked.
 [/INST] Note that 
$$111\cdots 1 = \frac{999\cdots 9}{9} = \frac{10^{91}-1}{9} =\frac{(10^{7})^{13}-1}{9} =\frac{ (10^{7}-1)((10^{7})^{12} + (10^{7})^{11} + \cdots + 1)}{9} $$
$$ = \frac{ 10^{7}-1}{9} ((10^{7})^{12} + (10^{7})^{11} + \cdots + 1) =1111111\cdot((10^{7})^{12} + (10^{7})^{11} + \cdots + 1)$$
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