Question

For any positive integer n that is larger than 2, find the largest common divisor for

n^5-5n^3+4n

Collected in the board: Number Theory

Steven Zheng posted 2 days ago

Answer

n^5-5n^3+4n

=n(n^4-5n^2+4)

=n(n^4-1-5n^2+5)

=n[(n^2-1)(n^2+1)-5(n^2-1)]

=n(n^2-1)(n^2-4)

=(n-2)(n-1)(n)(n+1)(n+2)

For any positive integer n that is larger than 2, the expression n^5-5n^3+4n has largest common divisor

1\times2\times3\times4\times5 = 120

Steven Zheng posted 2 days ago

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