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Which one of the following numbers is exactly divisible by $\left ( 11^{13} +1\right )$?

  1. $11^{26} +1$
  2. $11^{33} +1$
  3. $11^{39} -1$
  4. $11^{52} -1$
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We know that,

  • $x^{n} – y^{n}$ is divisible by $x+y,$ if $n$ is even.
  • $x^{n} – y^{n}$ is divisible by $x-y,$ if $n$ is odd.
  • $x^{n} + y^{n}$ is divisible by $x+y,$ if $n$ is odd.

 Now, we can check each option.

  1. $11^{26} + 1 = (11^{13})^{2} + 1^{2}$ is divisible by $11^{13}  + 1,$ if $2$ is odd. 
  2. $11^{33} + 1 = (11^{11})^{3} + 1^{3},$ here $11^{11} + 1 \neq 11^{13} + 1.$ 
  3. $11^{39} - 1 = (11^{13})^{3} - 1^{3},$ here $11^{13} - 1 \neq 11^{13} + 1.$ 
  4. $11^{52} - 1 = (11^{13})^{4} - 1^{4}$ is divisible by $11^{13}  + 1,$ if $4$ is even. 

So, the correct answer is $(D).$

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