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N has digits 1-7, not perfect square - 2011 Romania District VII p3

Source:

September 1, 2024

number theoryPerfect SquaresPerfect Square

Problem Statement

A positive integer

N

has the digits

1, 2, 3, 4, 5, 6

and

7

, so that each digit

i

i \in \{1, 2, 3, 4, 5, 6, 7\}

occurs

4i

times in the decimal representation of

N

. Prove that

N

is not a perfect square.