I recently attended a test which ask me two question based on perfect squares; here they are:
$1.$ How many even perfect squares between $1000$ and $5000$ are divisible by both $5$ and $9$?
$2.$ Can there be a perfect square whose digits consists of exactly $4$ ones, $4$ twos and $4$ zeroes in any order?
I haven't done anything much on perfect squares so I was reluctant to attempt them during exams, although as I had some spare time after finishing the other questions, I tried thinking about the first problem and I noticed that only $3600$ seems to satisfy the conditions, which is correct, but I don't know how to get this mathematically? And for the second problem, I don't have any clue till now.
Unfortunately they haven't given me any solutions for the questions, nor any ideas as to how to solve them (mathematically)?
Best Answer
For 1) Try to find all $k$ such that
$$ 1000 \leq 2\times 2 \times 5\times 5 \times 9 \times k^2 \leq 5000$$
For 2)
Any such number is divisible by $3$, but not by $9$ (look at the sum of digits, which gives the remainder upon dividing by $3$ and $9$).
In both, we are using the fact that if a prime $p$ divides $n^2$, then so does $p^2$.