[ Residue arithmetic (other).]  [ Divisibility rules for 3 and 9 ]

The numbers $1, 2, 3, \ldots, 1982$ are squared and written in sequence in some order.

Can the resulting multi-digit number be a perfect square?
When dividing the square of an integer by 3, the remainder is 0 or 1. The squares of the given numbers, when divided by 3, give remainders $1,1,0, \ldots, 1,1$. The sum of these remainders is $2 \cdot 1983: 3 \equiv 2(\bmod 3)$. The same remainder when divided by 3 is given by the number obtained by squaring the given numbers and writing them in any order.

## Answer

It cannot.