Problem 743

# Answer: 13

Solution. Multiply the numbers by 100, we get $\sqrt{73010}, \sqrt{73020}, \sqrt{73030}, \ldots, \sqrt{160030}$ (in this case, rational numbers will remain rational, and irrational numbers will remain irrational). The square root of a natural number $n$ is a rational number if and only if $n$ is a perfect square. Furthermore, a natural number ending in 0 (i.e., divisible by 2 and 5) can only be a perfect square if it ends in 00 (i.e., divisible by $2^{2}$ and $5^{2}$).

Thus, the required number is the number of perfect squares in the sequence of numbers $731, 732, 733, \ldots$, 1600. Since $27^{2}<731<28^{2}$ and $40^{2}=1600$, the answer is $40-27=13$.