GATE Electrical 2021 | GA Question: 4

Question

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

• A. $11^{26} +1$
• B. $11^{33} +1$
• C. $11^{39} -1$
• D. $11^{52} -1$

Answer 1

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.

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

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

Answer 2

11⁵² - 1

➠ (11²⁶)² - 1²

a² - b² = (a + b) (a - b)

➠ (11²⁶ + 1) (11²⁶ - 1)

➠ (11²⁶ + 1) [(11¹³)² - 1²]

➠ (11²⁶ + 1) (11¹³ + 1) (11¹³ - 1)

So , 11⁵² - 1 is exactly divisible by (11¹³ + 1)

I hope this helps you !