联立方程 ab=cd ，a+b+c+d=265 的正整数解有几组？

luyuanhong

如果限定 a≤b ，c≤d 和 a≤c ，则可以找到下列 52 组不同的正整数解：

( 1)   1*208 =   4 * 52
( 2)   2*156 =   3 *104
( 3)   2*204 =   8 * 51
( 4)   3*200 =  12 * 50
( 5)   4*153 =   6 *102
( 6)   4*196 =  16 * 49
( 7)   5*192 =  20 * 48
( 8)   6*150 =   9 *100
( 9)   6*188 =  24 * 47
(10)   7*184 =  28 * 46
(11)   8*147 =  12 * 98
(12)   8*180 =  32 * 45
(13)   9*176 =  36 * 44
(14)  10*144 =  15 * 96
(15)  10*172 =  40 * 43
(16)  11*168 =  42 * 44
(17)  12*141 =  18 * 94
(18)  12*164 =  41 * 48
(19)  13*160 =  40 * 52
(20)  14*138 =  21 * 92
(21)  14*156 =  39 * 56
(22)  15*152 =  38 * 60
(23)  16*135 =  24 * 90
(24)  16*148 =  37 * 64
(25)  17*144 =  36 * 68
(26)  18*132 =  27 * 88
(27)  18*140 =  35 * 72
(28)  19*136 =  34 * 76
(29)  20*129 =  30 * 86
(30)  20*132 =  33 * 80
(31)  21*128 =  32 * 84
(32)  22*124 =  31 * 88
(33)  22*126 =  33 * 84
(34)  23*120 =  30 * 92
(35)  24*116 =  29 * 96
(36)  24*123 =  36 * 82
(37)  25*112 = 100 * 28
(38)  26*108 =  27 *104
(39)  26*120 =  39 * 80
(40)  28*117 =  42 * 78
(41)  30*114 =  45 * 76
(42)  32*111 =  48 * 74
(43)  34*108 =  51 * 72
(44)  36*105 =  54 * 70
(45)  38*102 =  57 * 68
(46)  40* 99 =  60 * 66
(47)  42* 96 =  63 * 64
(48)  44* 93 =  62 * 66
(49)  46* 90 =  60 * 69
(50)  48* 87 =  58 * 72
(51)  50* 84 =  56 * 75
(52)  52* 81 =  54 * 78

但是，实际上，a 与 b 可以交换位置，c 与 d 可以交换位置，ab 与 cd 也可以交换位置，

所以上述 52 组解还要乘以 2×2×2=8 倍，本题共有 52×8=416 组正整数解。

天山草@

In[2]:= Solve[{a + b + c + d == 265, a b == c d, 0 < a, 0 < b, 0 < c,
  0 < d}, {a, b, c, d}, Integers]

Out[2]= {{a->1,b->208,c->4,d->52},{a->1,b->208,c->52,d->4},{a->2,b->156,c->3,d->104},{a->2,b->156,c->104,d->3},{a->2,b->204,c->8,d->51},{a->2,b->204,c->51,d->8},{a->3,b->104,c->2,d->156},{a->3,b->104,c->156,d->2},{a->3,b->200,c->12,d->50},{a->3,b->200,c->50,d->12},{a->4,b->52,c->1,d->208},{a->4,b->52,c->208,d->1},{a->4,b->153,c->6,d->102},{a->4,b->153,c->102,d->6},{a->4,b->196,c->16,d->49},{a->4,b->196,c->49,d->16},{a->5,b->192,c->20,d->48},{a->5,b->192,c->48,d->20},{a->6,b->102,c->4,d->153},{a->6,b->102,c->153,d->4},{a->6,b->150,c->9,d->100},{a->6,b->150,c->100,d->9},{a->6,b->188,c->24,d->47},{a->6,b->188,c->47,d->24},{a->7,b->184,c->28,d->46},{a->7,b->184,c->46,d->28},{a->8,b->51,c->2,d->204},{a->8,b->51,c->204,d->2},{a->8,b->147,c->12,d->98},{a->8,b->147,c->98,d->12},{a->8,b->180,c->32,d->45},{a->8,b->180,c->45,d->32},{a->9,b->100,c->6,d->150},{a->9,b->100,c->150,d->6},{a->9,b->176,c->36,d->44},{a->9,b->176,c->44,d->36},{a->10,b->144,c->15,d->96},{a->10,b->144,c->96,d->15},{a->10,b->172,c->40,d->43},{a->10,b->172,c->43,d->40},{a->11,b->168,c->42,d->44},{a->11,b->168,c->44,d->42},{a->12,b->50,c->3,d->200},{a->12,b->50,c->200,d->3},{a->12,b->98,c->8,d->147},{a->12,b->98,c->147,d->8},{a->12,b->141,c->18,d->94},{a->12,b->141,c->94,d->18},{a->12,b->164,c->41,d->48},{a->12,b->164,c->48,d->41},{a->13,b->160,c->40,d->52},{a->13,b->160,c->52,d->40},{a->14,b->138,c->21,d->92},{a->14,b->138,c->92,d->21},{a->14,b->156,c->39,d->56},{a->14,b->156,c->56,d->39},{a->15,b->96,c->10,d->144},{a->15,b->96,c->144,d->10},{a->15,b->152,c->38,d->60},{a->15,b->152,c->60,d->38},{a->16,b->49,c->4,d->196},{a->16,b->49,c->196,d->4},{a->16,b->135,c->24,d->90},{a->16,b->135,c->90,d->24},{a->16,b->148,c->37,d->64},{a->16,b->148,c->64,d->37},{a->17,b->144,c->36,d->68},{a->17,b->144,c->68,d->36},{a->18,b->94,c->12,d->141},{a->18,b->94,c->141,d->12},{a->18,b->132,c->27,d->88},{a->18,b->132,c->88,d->27},{a->18,b->140,c->35,d->72},{a->18,b->140,c->72,d->35},{a->19,b->136,c->34,d->76},{a->19,b->136,c->76,d->34},{a->20,b->48,c->5,d->192},{a->20,b->48,c->192,d->5},{a->20,b->129,c->30,d->86},{a->20,b->129,c->86,d->30},{a->20,b->132,c->33,d->80},{a->20,b->132,c->80,d->33},{a->21,b->92,c->14,d->138},{a->21,b->92,c->138,d->14},{a->21,b->128,c->32,d->84},{a->21,b->128,c->84,d->32},{a->22,b->124,c->31,d->88},{a->22,b->124,c->88,d->31},{a->22,b->126,c->33,d->84},{a->22,b->126,c->84,d->33},{a->23,b->120,c->30,d->92},{a->23,b->120,c->92,d->30},{a->24,b->47,c->6,d->188},{a->24,b->47,c->188,d->6},{a->24,b->90,c->16,d->135},{a->24,b->90,c->135,d->16},{a->24,b->116,c->29,d->96},{a->24,b->116,c->96,d->29},{a->24,b->123,c->36,d->82},{a->24,b->123,c->82,d->36},{a->25,b->112,c->28,d->100},{a->25,b->112,c->100,d->28},{a->26,b->108,c->27,d->104},{a->26,b->108,c->104,d->27},{a->26,b->120,c->39,d->80},{a->26,b->120,c->80,d->39},{a->27,b->88,c->18,d->132},{a->27,b->88,c->132,d->18},{a->27,b->104,c->26,d->108},{a->27,b->104,c->108,d->26},{a->28,b->46,c->7,d->184},{a->28,b->46,c->184,d->7},{a->28,b->100,c->25,d->112},{a->28,b->100,c->112,d->25},{a->28,b->117,c->42,d->78},{a->28,b->117,c->78,d->42},{a->29,b->96,c->24,d->116},{a->29,b->96,c->116,d->24},{a->30,b->86,c->20,d->129},{a->30,b->86,c->129,d->20},{a->30,b->92,c->23,d->120},{a->30,b->92,c->120,d->23},{a->30,b->114,c->45,d->76},{a->30,b->114,c->76,d->45},{a->31,b->88,c->22,d->124},{a->31,b->88,c->124,d->22},{a->32,b->45,c->8,d->180},{a->32,b->45,c->180,d->8},{a->32,b->84,c->21,d->128},{a->32,b->84,c->128,d->21},{a->32,b->111,c->48,d->74},{a->32,b->111,c->74,d->48},{a->33,b->80,c->20,d->132},{a->33,b->80,c->132,d->20},{a->33,b->84,c->22,d->126},{a->33,b->84,c->126,d->22},{a->34,b->76,c->19,d->136},{a->34,b->76,c->136,d->19},{a->34,b->108,c->51,d->72},{a->34,b->108,c->72,d->51},{a->35,b->72,c->18,d->140},{a->35,b->72,c->140,d->18},{a->36,b->44,c->9,d->176},{a->36,b->44,c->176,d->9},{a->36,b->68,c->17,d->144},{a->36,b->68,c->144,d->17},{a->36,b->82,c->24,d->123},{a->36,b->82,c->123,d->24},{a->36,b->105,c->54,d->70},{a->36,b->105,c->70,d->54},{a->37,b->64,c->16,d->148},{a->37,b->64,c->148,d->16},{a->38,b->60,c->15,d->152},{a->38,b->60,c->152,d->15},{a->38,b->102,c->57,d->68},{a->38,b->102,c->68,d->57},{a->39,b->56,c->14,d->156},{a->39,b->56,c->156,d->14},{a->39,b->80,c->26,d->120},{a->39,b->80,c->120,d->26},{a->40,b->43,c->10,d->172},{a->40,b->43,c->172,d->10},{a->40,b->52,c->13,d->160},{a->40,b->52,c->160,d->13},{a->40,b->99,c->60,d->66},{a->40,b->99,c->66,d->60},{a->41,b->48,c->12,d->164},{a->41,b->48,c->164,d->12},{a->42,b->44,c->11,d->168},{a->42,b->44,c->168,d->11},{a->42,b->78,c->28,d->117},{a->42,b->78,c->117,d->28},{a->42,b->96,c->63,d->64},{a->42,b->96,c->64,d->63},{a->43,b->40,c->10,d->172},{a->43,b->40,c->172,d->10},{a->44,b->36,c->9,d->176},{a->44,b->36,c->176,d->9},{a->44,b->42,c->11,d->168},{a->44,b->42,c->168,d->11},{a->44,b->93,c->62,d->66},{a->44,b->93,c->66,d->62},{a->45,b->32,c->8,d->180},{a->45,b->32,c->180,d->8},{a->45,b->76,c->30,d->114},{a->45,b->76,c->114,d->30},{a->46,b->28,c->7,d->184},{a->46,b->28,c->184,d->7},{a->46,b->90,c->60,d->69},{a->46,b->90,c->69,d->60},{a->47,b->24,c->6,d->188},{a->47,b->24,c->188,d->6},{a->48,b->20,c->5,d->192},{a->48,b->20,c->192,d->5},{a->48,b->41,c->12,d->164},{a->48,b->41,c->164,d->12},{a->48,b->74,c->32,d->111},{a->48,b->74,c->111,d->32},{a->48,b->87,c->58,d->72},{a->48,b->87,c->72,d->58},{a->49,b->16,c->4,d->196},{a->49,b->16,c->196,d->4},{a->50,b->12,c->3,d->200},{a->50,b->12,c->200,d->3},{a->50,b->84,c->56,d->75},{a->50,b->84,c->75,d->56},{a->51,b->8,c->2,d->204},{a->51,b->8,c->204,d->2},{a->51,b->72,c->34,d->108},{a->51,b->72,c->108,d->34},{a->52,b->4,c->1,d->208},{a->52,b->4,c->208,d->1},{a->52,b->40,c->13,d->160},{a->52,b->40,c->160,d->13},{a->52,b->81,c->54,d->78},{a->52,b->81,c->78,d->54},{a->54,b->70,c->36,d->105},{a->54,b->70,c->105,d->36},{a->54,b->78,c->52,d->81},{a->54,b->78,c->81,d->52},{a->56,b->39,c->14,d->156},{a->56,b->39,c->156,d->14},{a->56,b->75,c->50,d->84},{a->56,b->75,c->84,d->50},{a->57,b->68,c->38,d->102},{a->57,b->68,c->102,d->38},{a->58,b->72,c->48,d->87},{a->58,b->72,c->87,d->48},{a->60,b->38,c->15,d->152},{a->60,b->38,c->152,d->15},{a->60,b->66,c->40,d->99},{a->60,b->66,c->99,d->40},{a->60,b->69,c->46,d->90},{a->60,b->69,c->90,d->46},{a->62,b->66,c->44,d->93},{a->62,b->66,c->93,d->44},{a->63,b->64,c->42,d->96},{a->63,b->64,c->96,d->42},{a->64,b->37,c->16,d->148},{a->64,b->37,c->148,d->16},{a->64,b->63,c->42,d->96},{a->64,b->63,c->96,d->42},{a->66,b->60,c->40,d->99},{a->66,b->60,c->99,d->40},{a->66,b->62,c->44,d->93},{a->66,b->62,c->93,d->44},{a->68,b->36,c->17,d->144},{a->68,b->36,c->144,d->17},{a->68,b->57,c->38,d->102},{a->68,b->57,c->102,d->38},{a->69,b->60,c->46,d->90},{a->69,b->60,c->90,d->46},{a->70,b->54,c->36,d->105},{a->70,b->54,c->105,d->36},{a->72,b->35,c->18,d->140},{a->72,b->35,c->140,d->18},{a->72,b->51,c->34,d->108},{a->72,b->51,c->108,d->34},{a->72,b->58,c->48,d->87},{a->72,b->58,c->87,d->48},{a->74,b->48,c->32,d->111},{a->74,b->48,c->111,d->32},{a->75,b->56,c->50,d->84},{a->75,b->56,c->84,d->50},{a->76,b->34,c->19,d->136},{a->76,b->34,c->136,d->19},{a->76,b->45,c->30,d->114},{a->76,b->45,c->114,d->30},{a->78,b->42,c->28,d->117},{a->78,b->42,c->117,d->28},{a->78,b->54,c->52,d->81},{a->78,b->54,c->81,d->52},{a->80,b->33,c->20,d->132},{a->80,b->33,c->132,d->20},{a->80,b->39,c->26,d->120},{a->80,b->39,c->120,d->26},{a->81,b->52,c->54,d->78},{a->81,b->52,c->78,d->54},{a->82,b->36,c->24,d->123},{a->82,b->36,c->123,d->24},{a->84,b->32,c->21,d->128},{a->84,b->32,c->128,d->21},{a->84,b->33,c->22,d->126},{a->84,b->33,c->126,d->22},{a->84,b->50,c->56,d->75},{a->84,b->50,c->75,d->56},{a->86,b->30,c->20,d->129},{a->86,b->30,c->129,d->20},{a->87,b->48,c->58,d->72},{a->87,b->48,c->72,d->58},{a->88,b->27,c->18,d->132},{a->88,b->27,c->132,d->18},{a->88,b->31,c->22,d->124},{a->88,b->31,c->124,d->22},{a->90,b->24,c->16,d->135},{a->90,b->24,c->135,d->16},{a->90,b->46,c->60,d->69},{a->90,b->46,c->69,d->60},{a->92,b->21,c->14,d->138},{a->92,b->21,c->138,d->14},{a->92,b->30,c->23,d->120},{a->92,b->30,c->120,d->23},{a->93,b->44,c->62,d->66},{a->93,b->44,c->66,d->62},{a->94,b->18,c->12,d->141},{a->94,b->18,c->141,d->12},{a->96,b->15,c->10,d->144},{a->96,b->15,c->144,d->10},{a->96,b->29,c->24,d->116},{a->96,b->29,c->116,d->24},{a->96,b->42,c->63,d->64},{a->96,b->42,c->64,d->63},{a->98,b->12,c->8,d->147},{a->98,b->12,c->147,d->8},{a->99,b->40,c->60,d->66},{a->99,b->40,c->66,d->60},{a->100,b->9,c->6,d->150},{a->100,b->9,c->150,d->6},{a->100,b->28,c->25,d->112},{a->100,b->28,c->112,d->25},{a->102,b->6,c->4,d->153},{a->102,b->6,c->153,d->4},{a->102,b->38,c->57,d->68},{a->102,b->38,c->68,d->57},{a->104,b->3,c->2,d->156},{a->104,b->3,c->156,d->2},{a->104,b->27,c->26,d->108},{a->104,b->27,c->108,d->26},{a->105,b->36,c->54,d->70},{a->105,b->36,c->70,d->54},{a->108,b->26,c->27,d->104},{a->108,b->26,c->104,d->27},{a->108,b->34,c->51,d->72},{a->108,b->34,c->72,d->51},{a->111,b->32,c->48,d->74},{a->111,b->32,c->74,d->48},{a->112,b->25,c->28,d->100},{a->112,b->25,c->100,d->28},{a->114,b->30,c->45,d->76},{a->114,b->30,c->76,d->45},{a->116,b->24,c->29,d->96},{a->116,b->24,c->96,d->29},{a->117,b->28,c->42,d->78},{a->117,b->28,c->78,d->42},{a->120,b->23,c->30,d->92},{a->120,b->23,c->92,d->30},{a->120,b->26,c->39,d->80},{a->120,b->26,c->80,d->39},{a->123,b->24,c->36,d->82},{a->123,b->24,c->82,d->36},{a->124,b->22,c->31,d->88},{a->124,b->22,c->88,d->31},{a->126,b->22,c->33,d->84},{a->126,b->22,c->84,d->33},{a->128,b->21,c->32,d->84},{a->128,b->21,c->84,d->32},{a->129,b->20,c->30,d->86},{a->129,b->20,c->86,d->30},{a->132,b->18,c->27,d->88},{a->132,b->18,c->88,d->27},{a->132,b->20,c->33,d->80},{a->132,b->20,c->80,d->33},{a->135,b->16,c->24,d->90},{a->135,b->16,c->90,d->24},{a->136,b->19,c->34,d->76},{a->136,b->19,c->76,d->34},{a->138,b->14,c->21,d->92},{a->138,b->14,c->92,d->21},{a->140,b->18,c->35,d->72},{a->140,b->18,c->72,d->35},{a->141,b->12,c->18,d->94},{a->141,b->12,c->94,d->18},{a->144,b->10,c->15,d->96},{a->144,b->10,c->96,d->15},{a->144,b->17,c->36,d->68},{a->144,b->17,c->68,d->36},{a->147,b->8,c->12,d->98},{a->147,b->8,c->98,d->12},{a->148,b->16,c->37,d->64},{a->148,b->16,c->64,d->37},{a->150,b->6,c->9,d->100},{a->150,b->6,c->100,d->9},{a->152,b->15,c->38,d->60},{a->152,b->15,c->60,d->38},{a->153,b->4,c->6,d->102},{a->153,b->4,c->102,d->6},{a->156,b->2,c->3,d->104},{a->156,b->2,c->104,d->3},{a->156,b->14,c->39,d->56},{a->156,b->14,c->56,d->39},{a->160,b->13,c->40,d->52},{a->160,b->13,c->52,d->40},{a->164,b->12,c->41,d->48},{a->164,b->12,c->48,d->41},{a->168,b->11,c->42,d->44},{a->168,b->11,c->44,d->42},{a->172,b->10,c->40,d->43},{a->172,b->10,c->43,d->40},{a->176,b->9,c->36,d->44},{a->176,b->9,c->44,d->36},{a->180,b->8,c->32,d->45},{a->180,b->8,c->45,d->32},{a->184,b->7,c->28,d->46},{a->184,b->7,c->46,d->28},{a->188,b->6,c->24,d->47},{a->188,b->6,c->47,d->24},{a->192,b->5,c->20,d->48},{a->192,b->5,c->48,d->20},{a->196,b->4,c->16,d->49},{a->196,b->4,c->49,d->16},{a->200,b->3,c->12,d->50},{a->200,b->3,c->50,d->12},{a->204,b->2,c->8,d->51},{a->204,b->2,c->51,d->8},{a->208,b->1,c->4,d->52},{a->208,b->1,c->52,d->4}}
共有 104*4 = 416 组解。

luyuanhong

谢谢楼上 天山草 的解答。

luyuanhong

按照楼上的方法，找到的 52 组解为：

( 1)   1 * 208 =  52 *   4     ( 2)   3 * 104 = 156 *   2
( 3)   3 * 200 =  50 *  12     ( 4)   9 * 100 = 150 *   6
( 5)   5 * 192 =  48 *  20     ( 6)  15 *  96 = 144 *  10
( 7)   7 * 184 =  46 *  28     ( 8)  21 *  92 = 138 *  14
( 9)   9 * 176 =  44 *  36     (10)  27 *  88 = 132 *  18
(11)  11 * 168 =  42 *  44     (12)  33 *  84 = 126 *  22
(13)  13 * 160 =  40 *  52     (14)  39 *  80 = 120 *  26
(15)  15 * 152 =  38 *  60     (16)  45 *  76 = 114 *  30
(17)  17 * 144 =  36 *  68     (18)  51 *  72 = 108 *  34
(19)  19 * 136 =  34 *  76     (20)  57 *  68 = 102 *  38
(21)  21 * 128 =  32 *  84     (22)  63 *  64 =  96 *  42
(23)  23 * 120 =  30 *  92     (24)  69 *  60 =  90 *  46
(25)  25 * 112 =  28 * 100     (26)  75 *  56 =  84 *  50
(27)  27 * 104 =  26 * 108     (28)  81 *  52 =  78 *  54
(29)  29 *  96 =  24 * 116     (30)  87 *  48 =  72 *  58
(31)  31 *  88 =  22 * 124     (32)  93 *  44 =  66 *  62
(33)  33 *  80 =  20 * 132     (34)  99 *  40 =  60 *  66
(35)  35 *  72 =  18 * 140     (36) 105 *  36 =  54 *  70
(37)  37 *  64 =  16 * 148     (38) 111 *  32 =  48 *  74
(39)  39 *  56 =  14 * 156     (40) 117 *  28 =  42 *  78
(41)  41 *  48 =  12 * 164     (42) 123 *  24 =  36 *  82
(43)  43 *  40 =  10 * 172     (44) 129 *  20 =  30 *  86
(45)  45 *  32 =   8 * 180     (46) 135 *  16 =  24 *  90
(47)  47 *  24 =   6 * 188     (48) 141 *  12 =  18 *  94
(49)  49 *  16 =   4 * 196     (50) 147 *   8 =  12 *  98
(51)  51 *   8 =   2 * 204     (52) 153 *   4 =   6 * 102

2268493827

老罗的眼光真不行啊。哎

2268493827

数字游戏玩这么嗨。实际问题是个菜