Divisible by 169

[Moon]

Prove that $169 | 3^{3n+3}−26n−27$

[Avik Roy]

didn't try, but induction should work

[Zzzz]

Induction worked
It is easy to prove for n=1.

Let $169|3^{3m+1}-26m-27$, We need to prove that $169|3^{3(m+1)+1}-26(m+1)-27$
Now, \[169|3^{3m+1}-26m-27\]
\[\Rightarrow 3^{3m+1}\equiv 26m+27\ (mod\ 169)\]
\[\Rightarrow 27(3^{3m+1})\equiv 27(26m+27)\ (mod\ 169)\]
\[\Rightarrow 3^{3(m+1)+1}\equiv 27\cdot 26m +729\ (mod\ 169)\]
\[\Rightarrow 3^{3(m+1)+1}\equiv 26\cdot 26m+26m+676+53\ (mod\ 169)\]
\[\Rightarrow 3^{3(m+1)+1}\equiv 26m+53+169(4m+4)\ (mod\ 169)\]
\[\Rightarrow 3^{3(m+1)+1}\equiv 26m+53\ (mod\ 169)\]
\[\Rightarrow 3^{3(m+1)+1}\equiv 26(m+1)+27\ (mod\ 169)\]
\[\therefore 169|3^{3(m+1)+1}-26(m+1)-27\]

[Tahmid Hasan]

zzzz bhai apnei to kore dilen 1tu hide dile hoto na?

[Zzzz]

এটা অনেক দিন ধরে পড়ে আছে... সরাসরি করেই দিলাম

[Moon]

ঐ জু্বায়ের....পড়তে বস!!!

[Zzzz]

মন বসে না পড়ার টেবিলে

আর বইলেন না ভাই, পড়তে পড়তে অবস্থা খারাপ