For example, such numbers are $n_1 = 10^{100} - 1 = 99\ldots9$ and $n_2 = \frac{10^{100}}{2} - 1 = 49\ldots9$. Indeed, the numbers
$$n_1^3 - n_1 = (n_1 + 1)n_1(n_1 - 1) = 10^{100} \cdot n_1(n_1 - 1)$$
and
$$n_2^3 - n_2 = (n_2 + 1)n_2(n_2 - 1) = 10^{100} \cdot n_2 \cdot \frac{n_2 - 1}{2}$$
are divisible by $10^{100}$; this means that $n_i^3$ ends with $n_i$. On the other hand, the numbers $n_1^3 - n_1 = n_1(n_1 - 1)$ and $n_2^3 - n_2 = n_2(n_2 - 1)$ are not divisible by 5 (and even less by $10^{100}$); therefore, $n_i^2$ does not end with $n_i$.