question

\((r+2)^2\equiv (r+1)^2 \pmod{55}\\ r+2\equiv r+1\pmod{55}\)

what?????

someone help!

You have made a small mistake Max :)

\((r+2)^2=(r+1)^2\\ r+2=\pm (r+1)\\ \text{clearly r+2 can not equal r+1}\\ r+2=-(r+1)\\ r+2=-r-1\\ 2r=-3\\ r=-1.5\)

It would have been easier just to solve it as it was.

\(r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}\\ 2r \equiv-3 \pmod{55}\\ r \equiv -1.5 \pmod{55}\)

The solution of r should be a 4-digit solution, as the question requires......

The smallest 4-digit solution is when:

r=81, so the LHS of the equation is:

6,889 mod 55 = 14, and the RHS of the equation is:

6,724 mod 55 = 14.