7.3. In the notation of a natural number, there are 300 units, and the other digits are zeros. Can this number be a square of an integer?

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# Solution.

If the natural number $n$ were the square of an integer $a$, then from the condition it would follow that $n$ is divisible by 3, since the sum of the digits of this number is 300 and is divisible by 3. But then $a^2$ is also divisible by 3, meaning that $a^2 = n$ is divisible by 9, which cannot be the case, since the sum of the digits, which is 300, is not divisible by 9.

Answer: no.