[讨论]试证对任意正整数 n, [(n-1)! /(n +n^2)] 是偶数

ataorj

比如n=14时,(n-1)!/(n (1+n))是2,3,4,5,6,8,9,10,11,12,13的倍数.
[注]不是7的倍数.
In[5]:=
n=14;a=(n-1)!/(n (1+n))
Out[5]=
29652480
In[6]:=
Divisors[a]
Out[6]=
{1,2,3,4,5,6,8,9,10,11,12,13,15,16,18,20,22,24,26,27,30,32,33,36,39,40,44,45,\
48,52,54,55,60,64,65,66,72,78,80,81,88,90,96,99,104,108,110,117,120,128,130,\
132,135,143,144,156,160,162,165,176,180,192,195,198,208,216,220,234,240,256,\
260,264,270,286,288,297,312,320,324,330,351,352,360,384,390,396,405,416,429,\
432,440,468,480,495,512,520,528,540,572,576,585,594,624,640,648,660,702,704,\
715,720,768,780,792,810,832,858,864,880,891,936,960,990,1040,1053,1056,1080,\
1144,1152,1170,1188,1248,1280,1287,1296,1320,1404,1408,1430,1440,1485,1536,\
1560,1584,1620,1664,1716,1728,1755,1760,1782,1872,1920,1980,2080,2106,2112,\
2145,2160,2288,2304,2340,2376,2496,2560,2574,2592,2640,2808,2816,2860,2880,\
2970,3120,3168,3240,3328,3432,3456,3510,3520,3564,3744,3840,3861,3960,4160,\
4212,4224,4290,4320,4455,4576,4608,4680,4752,4992,5148,5184,5265,5280,5616,\
5632,5720,5760,5940,6240,6336,6435,6480,6656,6864,6912,7020,7040,7128,7488,\
7680,7722,7920,8320,8424,8448,8580,8640,8910,9152,9360,9504,9984,10296,10368,\
10530,10560,11232,11440,11520,11583,11880,12480,12672,12870,12960,13728,13824,\
14040,14080,14256,14976,15444,15840,16640,16848,16896,17160,17280,17820,18304,\
18720,19008,19305,19968,20592,20736,21060,21120,22464,22880,23040,23166,23760,\
24960,25344,25740,25920,27456,28080,28160,28512,29952,30888,31680,33280,33696,\
34320,34560,35640,36608,37440,38016,38610,41184,41472,42120,42240,44928,45760,\
46332,47520,49920,50688,51480,51840,54912,56160,57024,57915,59904,61776,63360,\
67392,68640,69120,71280,73216,74880,76032,77220,82368,84240,84480,89856,91520,\
92664,95040,99840,102960,103680,109824,112320,114048,115830,123552,126720,\
134784,137280,142560,149760,152064,154440,164736,168480,179712,183040,185328,\
190080,205920,207360,219648,224640,228096,231660,247104,253440,269568,274560,\
285120,299520,308880,329472,336960,366080,370656,380160,411840,449280,456192,\
463320,494208,539136,549120,570240,617760,658944,673920,741312,760320,823680,\
898560,926640,988416,1098240,1140480,1235520,1347840,1482624,1647360,1853280,\
1976832,2280960,2471040,2695680,2965248,3294720,3706560,4942080,5930496,\
7413120,9884160,14826240,29652480}