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

Question

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

Uniteasy Admin · Accepted 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
$$1	imes2	imes3	imes4	imes5 = 120$$

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