modular arithmeticcongruency

[main ruin]

t^2 - 15s +33u^2 -9r = 2
(t,s,u,r are in the set of integers)

Show this has no integer solutions using mod n

I used mod n to solve t^2 not being a solution however I am stumped at finding out why s u r are not solutions

[unique prism]

why did you calculate the remainder of t^2 modulo 3 for t going from 1 up to 12?

you just need to do that for t = 0, t=1 and t=2

other than this remark, your solution is already complete, you're just not fully understanding why for some reason

you can think like this:
Assume that there is a solution (t, s, u, r) over the integers.
Then t^2 == 2 (mod 3). But that is impossible.
Thus, this proves by contradiction that there is no solution to the equation, over Z.

[main ruin]

I forgot 🤣 + it’s a new concept to me so I’m extremely lost

And I realised this after

[main ruin]

Oh right cuz t doesn’t hold the equation won’t hold

Thanks

[unique prism]

i see, good : )