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Old 2007-07-08, 17:24   #2
VolMike
 
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Jun 2007
Moscow,Russia

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Well, I don't know the simplest way to reduce this problem, but it can be easy solved with one qiute obviously notice: any integer value can be represented as element of the set {15*k,15*k+1,15*k+2,...,15*k+14} for integer k=0,1,... Thus we can substitute this values for x in out initial expression and get results . Mathematica code, returns values of initial expression by substituting for x:
Code:
  (1/15) x (7 + 5 x^2 + 3 x^4) /. x -> 15*k + Range[0, 14] // Expand
And results:
Code:
  {7 k + 1125 k^3 + 151875 k^5,   1 + 37 k + 675 k^2 + 7875 k^3 + 50625 k^4 + 151875 k^5,   10 + 307 k + 4050 k^2 + 28125 k^3 + 101250 k^4 + 151875 k^5,   59 + 1357 k + 12825 k^2 + 61875 k^3 + 151875 k^4 + 151875 k^5,   228 + 4087 k + 29700 k^2 + 109125 k^3 + 202500 k^4 + 151875 k^5,   669 + 9757 k + 57375 k^2 + 169875 k^3 + 253125 k^4 + 151875 k^5,   1630 + 19987 k + 98550 k^2 + 244125 k^3 + 303750 k^4 + 151875 k^5,   3479 + 36757 k + 155925 k^2 + 331875 k^3 + 354375 k^4 + 151875 k^5,   6728 + 62407 k + 232200 k^2 + 433125 k^3 + 405000 k^4 + 151875 k^5,   12057 + 99637 k + 330075 k^2 + 547875 k^3 + 455625 k^4 + 151875 k^5,   20338 + 151507 k + 452250 k^2 + 676125 k^3 + 506250 k^4 + 151875 k^5,   32659 + 221437 k + 601425 k^2 + 817875 k^3 + 556875 k^4 +    151875 k^5,   50348 + 313207 k + 780300 k^2 + 973125 k^3 + 607500 k^4 + 151875 k^5,   74997 + 430957 k + 991575 k^2 + 1141875 k^3 + 658125 k^4 +    151875 k^5,   108486 + 579187 k + 1237950 k^2 + 1324125 k^3 + 708750 k^4 +    151875 k^5}
As can see, all expression values are integers for integer k. I think there is an shorter solution which uses the same idea, but represents in a brief way (expectedly with usage of properties of modulo operation).

Last fiddled with by VolMike on 2007-07-08 at 17:40
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