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GATE 2021 EE – Question 4

General Aptitude · Quantitative Aptitude: Arithmetic and Number Computation · 1 mark · Multiple choice

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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Correct answer: (D) $11^{52}-1$

Explanation

$11^{52}-1=\left(11^{26}-1\right)\left(11^{26}+1\right)$ and $11^{26}-1=\left(11^{13}-1\right)\left(11^{13}+1\right)$, so $11^{52}-1$ is divisible by $11^{13}+1$. Writing $a=11^{13}$, the other numbers are $a^2+1$, $a^{33/13}+1$ (not a power of $a$) and $a^3-1$, none of which has $a+1$ as a factor.