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Prove that $$\underbrace{11111\ldots1}_{91\text{ times}}$$ is a composite number and not a prime. Please give full steps of proving.

I tried and found that it is divisible by $1111111$ and $1111111111111$ but I can't prove it.

J. W. Tanner
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vishesh das
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  • Showing it is divisible by 1111111 proves it, how did you figure that out?. – marshal craft May 12 '17 at 10:06
  • Multiply 53 × 79 × 239 × 547 × 4649 × 14197 × 17837 × 4 262077 × 265 371653 × 43442 141653 × 316877 365766 624209 × 110 742186 470530 054291 318013 – Henry May 12 '17 at 10:09

2 Answers2

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$91=7\times13$. So the number is $\sum_{k=0}^{12}(1111111\times10^{7k})=1111111\sum_{k=0}^{12}10^{7k}$.

John Bentin
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Since $7|91$ we see that we can write $1111111$-($7\;$ $1$'s). Next to itself $13$ times and get the result of $111...$ (-$91$ times.)
This is equivalent to saying that $$1111111+(10^7\times 1111111) + (10^{14}\times 1111111) + ... + (10^{84}\times 1111111) = 1111... -91 \text{times}$$.

Which we can also see using a geometric series. Taking a factor of $1111111$ out of the LHS gives your result that $111111$ divides $1111...$(-$91$ times)

SEWillB
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