Each Digit Once in Powers of N

Number Theory Level 3

\(N\) is an integer. The numbers \(N^3\) and \(N^4\) together consist of all of the digits \(0,1,2,3,4,5,6,7,8,9\) exactly once. What is the value of \(N^2\)?

Details and assumptions

The number \(12 = 012\) does not contain the digit 0.

Each digit (0 to 9) appears in either \(N^3\) or \(N^4\) exactly once, and doesn't appear in both of them. For example, if \(N=2\), then \(N^3\) and \(N^4\) together consist of the digits \(1, 6, 8\) exactly once.


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