(Same unit and tens digits)

1. Determine the natural numbers $n$ such that $n$ and $n^{2}$ have the same unit digit.
2. Determine the natural numbers $n$ such that $n$ and $n^{2}$ have the same unit and tens digits.
1. We look modulo 10. We find (modulo 10): $0,1,5,62$. The candidates are those that end with the results from 1. It remains to check modulo 100. Let x be the tens digit.

$$
\begin{aligned}
& (10 x+a)^{2} \equiv 20 a x+a^{2} \quad(\bmod 100) \\
& \text { Hence }: 10 x+a \equiv 20 a x+a^{2} \quad(\bmod 100) \\
& (2 a-1) * 10 x \equiv a-a^{2} \quad(\bmod 100) \\
& \text { For } \mathrm{a}=0: x=0 \\
& \text { For } \mathrm{a}=1: x=0 \\
& \text { For } \mathrm{a}=5: 9 * 10 x=80 \quad(\bmod 100) \\
& \text { Therefore } 9 x=8 \quad(\bmod 10) \\
& 9 \equiv-1 \text { so it is its own inverse, thus } \\
& x \equiv 72 \equiv 2 \quad(\bmod 10) \\
& \text { For } \mathrm{a}=6: \\
& 10 x \equiv 70 \quad(\bmod 100) \\
& \text { hence } \mathrm{x}=7
\end{aligned}
$$

Finally, we get: $00,01,25,76$