Induction with divisibility

https://cdn.discordapp.com/attachments/322819040059850757/374672663706075156/unknown.png

I'm not quite sure why how you can stick an x behind the 11, can someone explain?

The x means that it is a multiple of 11.

so in otherwords it's 11 multiplied by a multiple of 11

so for example it could be that x is 121 so then $11x=11\cdot 121$ ? right

No its 11 multiplied by an integer. So x is just an integer.

So it could be 11*2 or 11*3 or 11*12 etc.

but why do i stick an integer there though?

https://cdn.discordapp.com/attachments/322819040059850757/374672663706075156/unknown.png

I'm not quite sure why how you can stick an x behind the 11, can someone explain?

Well this notation is unfamiliar to me however i assume that it means that 5^(2n)-3^(n) is divisible by 11 for all real values of x. So when u divide 5^(2n)-3^(n) by 11 u will always get an integer value which they have denoted with ‘x’.

where is this part, i can only see n in the set of naturals in the beginning and p(n) implies there exists a z in the set of integers


anyway you always get an integer? how tho but can't you get a fraction out?

Huh? I struggle to find what you're pointing at to be honest...

If you mean the part where it says $5^{2n}-3^n=11x$ then of course you're allowed to stick an x there, in fact you need to, because you are assuming here that $5^{2n}-3^n$ is a multiple of 11, which serves as your induction assumption that is used later on.

Here $x$ is just some integer, depending on your choice of $n$ of course in your assumption. Missing out on an $x$ would mean that for any $n \in \mathbb{N}$ that I'd pick, I would end up with the number 11 from $5^{2n}-3^n$ which is counter intuitive at the very least.

Sorry mate im not familiar with all of this. I gave u an answer that i thought made sense based on my knowledge of induction, and i tried to extrapolate what i knew to fit what i saw.

Guess i am wrong or missing something.Sorry im not gonna be much help.