From question 1, i thought that fermat's little theorem. a^p≡a ≡b^p ≡ b (mod p )
and because (a,p)=1=(b,p) , (a,p^2)=1=(b,p^2), a^(p^2)≡a≡b≡b^(p^2) mod p^2 but how can we know that a^p≡b^p mod p^2 ???
from question 2, it is alike using chinese theorem. but is chinese theorem used in the same unknown..?? for example, x≡a mod m , x≡b mod n => x=c mod mn (unique, m,n are co prime.)
but in this question, .. i don't know well i think that the x, and y are not same unknown and also there can not be exist.. when x^≡a ... ( in chinese theorem, x was not quadratic..)