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Math Help - The largest number N

  1. #1
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    The largest number N

    Another question:

    What is the largest number N for which you can say that n^5-5n^3+4n is divisible by N for every integer n?

    Thank you very much.
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  2. #2
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    Hello, Jenny!

    What is the largest number N for which you can say that
    n^5-5n^3+4n is divisible by N for every integer n?

    \text{Factor: }\:n^5 - 5n^3 + 4n \;=\;n(n^4 - 4n^2 + 4)

    . . . . . . . . = \;n(n^2 - 1)(n^2 - 4)\;=\;n(n-1)(n+1)(n-2)(n+2)


    \text{We have: }\:n^5 - 5n^3 + 4n \;=\;\underbrace{(n-2)(n-1)(n)(n+1)(n-2)}_{\text{product of 5 consecutive integers}}

    The product of 5 consecutive integers is divisible by 1,\,2,\,3,\,4,\,5.

    Therefore: . N \,=\,120

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