$\displaystyle \begin{array}{{>{\displaystyle}l}} 56\\ \frac{9}{25} x^{2} +\frac{12}{35} xy+\frac{4}{49} y^{2}\\ =\left(\frac{3}{5} x\right)^{2} +2.\frac{3}{5} .\frac{2}{7} +\left(\frac{2}{7} y\right)^{2}\\ =\left(\frac{3}{5} x+\frac{2}{7} y\right)^{2}\\ Với\ x=5;y=-7\\ \left(\frac{3}{5} .5+\frac{2}{7} .-7\right)^{2}\\ =( 3-2)^{2} =1\ \\ 57\ \\ \frac{25}{16} u^{4} v^{2} +\frac{1}{5} u^{2} v^{3} +\frac{4}{625} v^{4} \ \\ =\left(\frac{5}{4} u^{2} v\right)^{2} +2.\frac{5}{4} u^{2} v.\frac{2}{25} +\left(\frac{2}{25} v^{2}\right)^{2}\\ \left(\frac{5}{4} u^{2} v+\frac{2}{25} v^{2}\right)^{2}\\ Với\ u=\frac{2}{5} ;v=-5\ \\ \rightarrow \left(\frac{5}{4} .\frac{4}{25} .-5+\frac{2}{25} .25\right)^{2}\\ ( -1+2)^{2} =1\ \\ 58\ \\ Ta\ có\ :\ n\ chia\ cho\ 11\ dư\ 4\ \\ \rightarrow n=11k+4\ \\ n^{2} =121k^{2} +88k+16\ \\ Vì\ 121\ chia\ hết\ cho\ 11\ \\ 88\ chia\ hết\ cho\ 11\ \\ 16\ chia\ cho\ 11\ dư\ 5\ \\ \rightarrow \ n^{2} \ chia\ cho\ 11\ dư\ 5\ \\ .59\ \\ Ta\ có\ :\ n\ chia\ cho\ 13\ dư\ 7\ \\ \rightarrow n=13k+7\ \\ n^{2} -10=169k^{2} +182k+49-10\ \\ n^{2} -10=169k^{2} +182k+39\ \\ Ta\ có\ :\ 169\ chia\ hết\ cho\ 13\ \\ 182\ chia\ hết\ cho\ 13\ \\ 39\ chia\ hết\ cho\ 13\ \\ do\ đó\ :\ n^{2} -10\ chia\ hết\ cho\ 13\ \end{array}$
