Question

\[\text{find }n\text{ for wich the expression}\\2^n+3^n \text{ is divisible by} 13\]

Answer 1

\[\large 13|2^n+3^n\\ 13|2^n ,13|3^n\]

Answer 2

That's not necessarily true. For instance, 2|(1+3) but 2 doesn't divide either 1 or 3.

Answer 3

so we need to have
\[\large 2^n\equiv 0\mod13\]

Answer 4

look at the question

Answer 5

we r looking for n that will make this true

Answer 6

@mukushla

Answer 7

2 wil make it true,
but 13 wont divide 2^2 or 3^2

Answer 8

i think your start is a bad one

Answer 9

but we r looking for a scenario where it divides them

Answer 10

okay number theory???

Answer 11

approach

Answer 12

n = 2
will make the equation 2^n + 3^n = 13k true right ?

i want u pause and check ur start

Answer 13

oh @mukushla is here :)

Answer 14

I found that n=2, 6, and 10 work, so a conjecture could be made that n=4k+2 where k is an integer will work.
Also, this is provable via induction on k.
Inductive step: Assume that \[2^{4k+2}+3^{4k+2}\equiv0\pmod{13}\]
We must prove that
\[2^{4(k+1)+2}+3^{4(k+1)+2}\equiv0\pmod{13}\]
To prove this, we have
\[\begin{align}
2^{4(k+1)+2}+3^{4(k+1)+2}&\equiv2^{4k+4+2}+3^{4k+4+2}\pmod{13}\\
&\equiv2^{4k+2}+3^{4k+2}+16+81\pmod{13}\\
&\equiv 2^{4k+2}+3^{4k+2}+97\pmod{13}\\
&\equiv 2^{4k+2}+3^{4k+2}\pmod{13}\text{ because }13|97\\
&\equiv 0\pmod{13}\text{ by inductive assumption}
\end{align}\]
Thus, if n=4k+2, where k is an integer, then 2^n+3^n will divide 13.

Answer 15

okay i see ,but is the any other numbers ...say less than hundred,\[1\len\le 100\]

Answer 16

\[1\le n \le100\]

Answer 17

Just plug in k=1, 2, ..., 24 (k=24 gives n=98) and you get all n where 13 divides 2^n+3^n, and n is between 1 and 100.

Answer 18

so a follow up question says.hence
prove that
\[13|2^{50}+3^{50}\]

Answer 19

thanks at @blockcolder for the first part

Answer 20

50=4(12)+2, so simply let k=12.

Answer 21

lets say we didnt do the induction how can we do this from scratch

Answer 22

wat steps are necesarry to show this

Answer 23

by fermat
\[2^{12}\equiv 1\mod 13\]

Answer 24

\[\large 2^{50}=2^{2(12)}2^2\]