7.1. In a row, the numbers are written: $100^{100}, 101^{101}, 102^{102}, \ldots, 234^{234}$ (i.e., the numbers of the form $n^{n}$ for natural numbers n from 100 to 234). How many of the listed numbers are perfect squares? (A perfect square is the square of an integer.)
# Answer: 71

Solution. Consider a number of the form $m^{k}$, where $m$ and $k$ are natural numbers. If $k$ is even, then $m^{k}$ is a perfect square. If $k$ is odd, then $m^{k}$ is a perfect square if and only if $m$ is a perfect square. Thus, the answer to our problem is the total number of even numbers and odd perfect squares in the set of numbers $\{100,101,102, \ldots, 234\}$. The number of even numbers needed is $(234-98) / 2=68$. The number of perfect squares needed is exactly $3: 11^{2}, 13^{2}, 15^{2}$. In total, there are $68+3=71$ numbers that meet the criteria.