[tex]Numarul~din~enunt~este~egal~cu: \\ \\ \underbrace{999...9}_{\mbox{n}}\underbrace{000...0}_{\mbox{n+2}}+25= \\ \\ =(10^{n}-1) \cdot 10^{n+2}+25= \\ \\ =10^{2n+2}-10^{n+2} +5^2= \\ \\ =(10^{n+1})^2-2 \cdot 5 \cdot 10^{n+1}+5^2= \\ \\ = (10^{n+1}+5)^2-patrat~perfect.[/tex]
[tex]------------------------------- \\ \\ Notam~\underbrace{111...1}_{\mbox{n-1}}=k. \\ \\ Observam~ca~10^{n-1}=\underbrace{999...9}_{\mbox{n-1}}+1=9k+1. \\ \\ Deci~a=(100k+11)k+9k+1= \\ \\ =100k^2+11k+9k+1= \\ \\ =100k^2+20k+1= \\ \\ =(10k+1)^2-patrat~perfect. [/tex]
