Understand the problem

Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.
Source of the problem
Centroamerican olympiad 2016
Topic
Number theory
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Note that the digits can only be 0,1,4,9. Also, the last digit has to be 0.
The sum of the digits has to be divisible by 3. Hence (checking by hand), the possible candidates are 1110,1140,1410,4410,4140,9000,9090,9900 and 9990. Note that the two initial examples were 1110 and 9000. The other ones came by replacing one or more the digits with other digits that are equivalent modulo 3.
Check for multiples of 7 in the list.
The answer is 4410.

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