質因子最多只含2和5的數

性質

分母為質因子最多只含2和5的數的分數必定是有限小數。

金字塔

1

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

2

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1

1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1

1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1

1 20 190 1140 4845 15504 38760 77520 125970 167960 184756 167960 125970 77250 38760 15504 4845 1140 190 20 1

1 21 210 1330 5985 20349 54264 116280 203490 293930 352716 352716 293930 203490 116280 54264 20349 5985 1330 210 21 1

1 22 231 1540 7315 26334 74613 170544 319770 497420 646646 705432 646646 497420 319770 170544 74613 26334 7315 1540 231 22 1

1 23 253 1771 8855 33649 100947 245157 490314 817190 1144066 1352078 1352078 1144066 817190 490314 245157 100947 33649 8855 1771 253 23 1

1 24 276 2024 10626 42504 134596 346104 735471 1307504 1961256 2496144 2704156 2496144 1961256 1307504 735471 346104 134596 42504 10626 2024 276 24 1

1 25 300 2300 12650 53130 177100 480700 1081575 2042975 3268760 4457400 5200300 5200300 4457400 3268760 2042975 1081575 480700 177100 53130 12650 2300 300 25 1

1 26 325 2600 14950 65780 230230 657800 1562275 3124550 5311735 7726160 9657700 10400600 9657700 7726160 5311735 3124550 1562275 657800 230230 65780 14950 2600 325 26 1

3

1

5 5

25 50 25

125 375 375 125

625 2500 3750 2500 625

3125 15625 31250 31250 15625 3125

15625 93750 234375 312500 234375 93750 15625

78125 546875 1640625 2734375 2734375 1640625 546875 78125

390625 3125000 10937500 21875000 27343750 21875000 10937500 3125000 390625

1953125 17578125 70312500 164062500 234375000 234375000 164062500 70312500 17578125 1953125

9765625 97656250 439453125 1171875000 2050781250 2460937500 2050781250 1171875000 439453125 97656250 9765625

48828125 537109375 2685546875 8056640625 16113281250 22558593750 22558593750 16113281250 8056640625 2685546875 537109375 48828125

244140625 2929687500 16113281250 53710937500 120849609375 193359375000 225585937500 193359375000 120849609375 53710937500 16113281250 292968750 244140625

1220703125 15869140625 95214843750 349121093750 872802734375 1569824218750 2094726562500 2094726562500 1569824218750 872802734375 349121093750 95214843750 15869140625 1220703125

6103515625 85449218750 555419921875 2221679687500 6109619140625 12219238231250 18328857421875 20947656250000 18328857421875 12219238231250 6109619140625 2221679687500 555419921875 85449218750 6103515625

30517578125 457763671875 204345703125 13885498046875 41656494140625 91644287109375 152740478515625 196380615234375 196380615234375 152740478515625 91644287109375 41656494140625 13885498046875 3204345703125 457763671875 30517578125

1

0.5 0.5

0.25 0.5 0.25

0.125 0.375 0.375 0.125

0.0625 0.25 0.375 0.25 0.0625

0.03125 0.15625 0.3125 0.3125 0.15625 0.03125

0.015625 0.09375 0.234375 0.3125 0.234375 0.09375 0.015625

0.0078125 0.0546875 0.1646025 0.2734375 0.2734375 0.1646025 0.0546875 0.0078125

0.00390625 0.03125 0.109375 0.21875 0.2734375 0.21875 0.109375 0.03125 0.00390625

0.001953125 0.017578125 0.0703125 0.1640625 0.234375 0.234375 0.1640625 0.0703125 0.017578125 0.001953125

0.0009765625 0.009765625 0.0439453125 0.1171875 0.205078125 0.24609375 0.205078125 0.1171875 0.0439453125 0.009765625 0.0009765625

0.00048828125 0.00537109375 0.02685546875 0.08056640625 0.1611328125 0.2255859375 0.2255859375 0.1611328125 0.08056640625 0.02685546875 0.00537109375 0.00048828125

0.000244140625 0.0029296875 0.01611328125 0.0537109375 0.120849609375 0.193359375 0.2255859375 0.193359375 0.120849609375 0.0537109375 0.01611328125 0.0029296875 0.000244140625

0.0001220703125 0.0015869140625 0.009521484375 0.034912109375 0.0872802734375 0.156982421875 0.20947265625 0.20947265625 0.156982421875 0.0872802734375 0.034912109375 0.009521484375 0.0015869140625 0.0001220703125

0.00006103515625 0.0008544921875 0.00555419921875 0.022216796875 0.06109619140625 0.1221923823125 0.18328857421875 0.2094765625 0.18328857421875 0.1221923823125 0.06109619140625 0.022216796875 0.00555419921875 0.0008544921875 0.00006103515625

0.000030517578125 0.000457763671875 0.003204345703125 0.013885498046875 0.041656494140625 0.091644287109375 0.152740478515625 0.196380615234375 0.196380615234375 0.152740478515625 0.091644287109375 0.041656494140625 0.013885498046875 0.003204345703125 0.000457763671875 0.000030517578125

0.0000152587890625 0.000244140625 0.0018310546875 0.008544921875 0.01666259765625 0.111083984375 0.1221923828125 0.174560546875 0.196380615234375 0.174560546875 0.1221923828125 0.111083984375 0.01666259765625 0.008544921875 0.0018310546875 0.000244140625 0.0000152587890625

0.00000762939453125 0.00012969970703125 0.00103759765625 0.00518798828125 0.018157958984375 0.047210693359375 0.09442138671875 0.14837646484375 0.1854705810546875 0.1854705810546875 0.14837646484375 0.09442138671875 0.047210693359375 0.018157958984375 0.00518798828125 0.00103759765625 0.00012969970703125 0.00000762939453125

0.000003814697265625 0.00006866455078125 0.000583648681640625 0.00311279296875 0.0116729736328125 0.032684326171875 0.0708160400390625 0.12139892578125 0.16692352294921875 0.1854705810546875 0.16692352294921875 0.12139892578125 0.0708160400390625 0.032684326171875 0.0116729736328125 0.00311279296875 0.000583648681640625 0.00006866455078125 0.000003814697265625

0.0000019073486328125 0.0000362396240234375 0.0003261566162109375 0.0018482208251953125 0.00739288330078125 0.02217864990234375 0.05175018310546875 0.09610748291015625 0.144161224365234375 0.176197052001953125 0.176197052001953125 0.144161224365234375 0.09610748291015625 0.05175018310546875 0.02217864990234375 0.00739288330078125 0.0018482208251953125 0.0003261566162109375 0.0000362396240234375 0.0000019073486328125

4

1

2 1

4 4 1

8 12 6 1

16 32 24 8 1

32 80 80 40 10 1

64 192 240 160 60 12 1

128 448 672 560 280 84 14 1

256 1024 1792 1792 1120 448 112 16 1

512 2304 4608 5376 4032 2016 672 144 18 1

1024 5120 11520 15360 13440 8064 3360 960 180 20 1

2048 11264 28160 42240 42240 29568 14784 5280 1320 220 22 1

4096 24576 67584 112640 126720 101376 59136 25344 7920 1760 264 24 1

5

1

1 1 1

1 2 3 2 1

1 3 6 7 6 3 1

1 4 10 16 19 16 10 4 1

1 5 15 30 45 51 45 30 16 5 1

1 6 21 50 90 126 141 126 90 50 21 6 1

1 7 28 77 161 266 357 393 357 266 161 77 28 7 1

1 8 36 112 266 504 784 1016 1107 1016 784 504 266 112 36 8 1

1 9 45 156 414 882 1554 2304 2907 3139 2907 2304 1554 882 414 156 45 9 1

1 10 55 210 615 1452 2850 4740 6765 8350 8953 8350 6765 4740 2850 1452 615 210 55 10 1

1 11 66 275 880 2277 4917 9042 14355 19855 24068 25653 24068 19855 14355 9042 4917 2277 880 275 66 11 1

1 12 78 352 1221 3432 8074 16236 28314 43252 58278 69576 73789 69576 58278 43252 28314 16236 8074 3432 1221 352 78 12 1

1 13 91 442 1651 5005 12727 27742 52624 87802 129844 171106 201643 212941 201643 171106 129844 87802 52624 27742 12727 5005 1651 442 91 13 1

1 14 105 546 2184 7098 19383 45474 93093 168168 270270 388752 502593 585690 616227 585690 502593 388752 270270 168168 93093 45474 19383 7098 2184 546 105 14 1

6

1

1 1 1 1

1 2 3 4 3 2 1

1 3 6 10 12 12 10 6 3 1

1 4 10 20 31 40 44 40 31 20 10 4 1

1 5 15 35 65 101 135 155 155 135 101 65 35 15 5 1

1 6 21 56 120 216 336 456 546 580 546 456 336 216 120 56 21 6 1

1 7 28 84 203 413 728 1128 1554 1918 2128 2128 1918 1554 1128 728 413 203 84 28 7 1

1 8 36 120 322 728 1428 2472 3823 5328 6728 7728 8092 7728 6728 5328 3823 2472 1428 728 322 120 36 8 1

1 9 45 165 486 1206 2598 4950 8451 13051 18351 23607 27876 30276 30276 27876 23607 18351 13051 8451 4950 2598 1206 486 165 54 9 1

1 10 55 220 705 1902 4455 9240 17205 29050 44803 63460 82885 100110 112035 116304 112035 100110 82885 63460 44803 29050 17205 9240 4455 1902 705 220 55 10 1

質因子最多只含2和3的數的金字塔

1

2 3

4 6 9

8 12 18 27

16 24 36 54 81

32 48 72 108 162 243

64 96 144 216 324 486 729

128 192 288 432 648 972 1458 2187

256 384 576 864 1296 1944 2916 4374 6561

512 768 1152 1728 2592 3888 5832 8748 13122 19683

1024 1536 2304 3456 5184 7776 11664 17496 26244 39366 59049

2048 3072 4608 6912 10368 15552 23328 34992 52488 78732 118098 177147

4096 6144 9216 13824 20736 31104 46656 69984 104976 157464 236196 354294 531441

8192 12288 18432 27648 41472 62208 93312 139968 209952 314928 472392 708582 1062882 1594323

16384 24576 36864 55296 82944 124416 186624 279936 419904 629856 944784 1417164 2125764 3188646 4782969

32768 49152 73728 110592 165888 248832 373248 559872 839808 1259712 889568 2834352 4251528 6377292 9565938 14348907

65536 98304 147456 221184 331776 497664 746496 1119744 1679616 2519424 3779136 5668704 8503056 12754584 19131876 28697814 43046721

131072 196608 294912 442368 663552 995328 1492992 2239488 3359232 5038848 7558272 11337408 17006112 25509168 38263752 57395628 86093442 129140163

262144 393216 589824 884736 1327104 1990656 2985984 4478976 6718464 10077696 15116544 22674816 34012224 51018336 76527504 114791256 172186884 258280326 387420489

524288 786432 1179648 1769472 2654208 3981312 5971968 8957952 13436928 20155392 30233088 45349632 68024448 102036672 153055008 229582512 344373768 516560652 774840978 1162261467

質因子最多只含3和5的數的金字塔

1

3 5

9 15 25

27 45 75 125

81 135 225 375 625

243 405 675 1125 1875 3125

729 1215 2025 3375 5625 9375 15625

2187 3645 6075 10125 16875 28125 46875 78125

6561 10935 18225 30375 50625 84375 140625 234375 390625

19683 32805 54675 91125 151875 253125 421875 703125 1171875 1953125

59049 98415 164025 273375 455625 759375 1265625 2109375 3515625 5859375 9765625

177147 295245 492075 1090125 1366875 2278125 3796875 6328125 10546875 17578125 48828125

質因子最多只含2和5的數的金字塔

1

2 5

4 10 25

8 20 50 125

16 40 100 250 625

32 80 200 500 1250 3125

64 160 400 1000 2500 6250 15625

128 320 800 2000 5000 12500 31250 78125

256 640 1600 4000 10000 25000 62500 156250 390625

512 1280 3200 8000 20000 50000 125000 312500 781250 1953125

1024 2560 6400 16000 40000 100000 250000 625000 1562500 3906250 9765625

2048 5120 12800 32000 80000 200000 500000 1250000 3125000 7812500 19531250 48828125

4096 10240 25600 64000 160000 400000 1000000 2500000 6250000 15625000 39062500 97656250 244140625

8192 20480 51200 128000 320000 800000 2000000 5000000 12500000 31250000 78125000 195312500 488281250 1220703125

16384 40960 102400 256000 640000 1600000 4000000 10000000 25000000 62500000 156250000 390625000 976562500 2441406250 6103515625

32768 81920 204800 512000 1280000 3200000 8000000 20000000 50000000 125000000 312500000 781250000 1953125000 4882812500 12207031250 30517578125

65536 163840 409600 1024000 2560000 6400000 16000000 40000000 100000000 250000000 625000000 1562500000 3906250000 9765625000 24414062500 61035156250 152587890625

131072 327680 819200 2048000 5120000 12800000 32000000 80000000 200000000 500000000 1250000000 3125000000 7812500000 19531250000 48828125000 122070312500 305175781250 762939453125

262144 655360 1638400 4096000 10240000 25600000 64000000 160000000 400000000 1000000000 2500000000 6250000000 15625000000 39062500000 97656250000 244140625000 610351562500 1525878906250 3814697265625

524288 1310720 3276800 8192000 20480000 51200000 128000000 320000000 800000000 2000000000 5000000000 12500000000 31250000000 78125000000 195312500000 488281250000 1220703125000 3051757812500 7629394531250 19073486328125

1048576 2621440 6553600 16384000 40960000 102400000 256000000 640000000 1600000000 4000000000 10000000000 25000000000 62500000000 156250000000 390625000000 976562500000 2441406250000 6103515625000 15258789062500 38146972656250 95367431640625

2097152 5242880 13107200 32768000 81920000 204800000 512000000 1280000000 3200000000 8000000000 20000000000 50000000000 125000000000 312500000000 781250000000 1953125000000 4882812500000 12207031250000 30517578125000 76293945312500 190734863281250 476837158203125

4194304 10485760 26214400 65536000 163840000 409600000 1024000000 2560000000 6400000000 16000000000 40000000000 100000000000 250000000000 625000000000 1562500000000 3906250000000 9765625000000 24414062500000 61035156250000 152587890625000 381469726562500 953674316406250 2384185791015625

8388608 20971520 52428800 131072000 327680000 819200000 2048000000 5120000000 12800000000 32000000000 80000000000 200000000000 500000000000 1250000000000 3125000000000 7812500000000 19531250000000 48828125000000 122070312500000 305175781250000 762939453125000 1907348632812500 4768371582031250 11920928955078125

16777216 41943040 104857600 262144000 655360000 1638400000 4096000000 10240000000 25600000000 64000000000 160000000000 400000000000 1000000000000 2500000000000 6250000000000 15625000000000 39062500000000 976562500000000 244140625000000 610351562500000 1525878906250000 3814697265625000 9536743164062500 23841857910156250 59604644775390625

33554432 83886080 209715200 524288000 1310720000 3276800000 8192000000 20480000000 51200000000 128000000000 320000000000 800000000000 2000000000000 5000000000000 12500000000000 31250000000000 78125000000000 195312500000000 488281250000000 1220703125000000 3051757812500000 7629394531250000 19073486328125000 47683715820312500 119209289550781250 298023223876953125

67108864 167772160 419430400 1048576000 2621440000 6553600000 16384000000 40960000000 102400000000 256000000000 640000000000 1600000000000 4000000000000 10000000000000 25000000000000 62500000000000 156250000000000 390625000000000 976562500000000 2441406250000000 6103515625000000 15258789062500000 38146972656250000 95367431640625000 238418579101562500 596046447753906250 1490116119384765625

134217728 335544320 838860800 2097152000 5242880000 13107200000 32768000000 81920000000 204800000000 512000000000 1280000000000 3200000000000 8000000000000 20000000000000 50000000000000 125000000000000 312500000000000 781250000000000 1953125000000000 4882812500000000 12207031250000000 30517578125000000 76293945312500000 190734863281250000 476837158203125000 1192092895507812500 2980232238769531250 7450580596923828125

268435456 671088640 1677721600 4194304000 10485760000 26214400000 65536000000 163840000000 409600000000 1024000000000 2560000000000 6400000000000 16000000000000 40000000000000 100000000000000 250000000000000 625000000000000 1562500000000000 3906250000000000 9765625000000000 24414062500000000 61035156250000000 152587890625000000 381469726562500000 953674316406250000 2384185791015625000 5960464477539062500 14901161193847656250 37252902984619140625

536870912 1342177280 3355443200 8388608000 20971520000 52428800000 131072000000 327680000000 819200000000 2048000000000 5120000000000 12800000000000 32000000000000 80000000000000 200000000000000 500000000000000 1250000000000000 3125000000000000 7812500000000000 19531250000000000 48828125000000000 122070312500000000 305175781250000000 762939453125000000 1907348632812500000 4768371582031250000 11920928955078125000 29802322387695312500 74505805969238281250 186264514923095703125

1073741824 2684354560 6710886400 16777216000 41943040000 104857600000 262144000000 655360000000 1638400000000 4096000000000 10240000000000 25600000000000 64000000000000 160000000000000 400000000000000 1000000000000000 2500000000000000 6250000000000000 15625000000000000 39062500000000000 97656250000000000 244140625000000000 610351562500000000 1525878906250000000 3814697265625000000 9536743164062500000 23841857910156250000 59604644775390625000 149011611938476562500 372529029846191406250 931322574615478515625

2147483648 5368709120 13421772800 33554432000 83886080000 209715200000 524288000000 1310720000000 3276800000000 8192000000000 20480000000000 51200000000000 128000000000000 320000000000000 800000000000000 2000000000000000 5000000000000000 12500000000000000 31250000000000000 78125000000000000 195312500000000000 488281250000000000 1220703125000000000 3051757812500000000 7629394531250000000 19073486328125000000 47683715820312500000 119209289550781250000 298023223876953125000 745058059692382812500 1862645149230957031250 4656612873077392578125

4294967296 10737418240 26843545600 67108864000 167772160000 419430400000 1048576000000 2621440000000 6553600000000 16384000000000 40960000000000 102400000000000 256000000000000 640000000000000 1600000000000000 4000000000000000 10000000000000000 25000000000000000 62500000000000000 156250000000000000 390625000000000000 976562500000000000 2441406250000000000 6103515625000000000 15258789062500000000 38146972656250000000 95367431640625000000 238418579101562500000 596046447753906250000 1490116119384765625000 3725290298461914062500 9313225746154785156250 23283064365386962890625

8589934592 21474836480 53687091200 134217728000 335544320000 838860800000 2097152000000 5242880000000 13107200000000 32768000000000 81920000000000 204800000000000 512000000000000 1280000000000000 3200000000000000 8000000000000000 20000000000000000 50000000000000000 125000000000000000 312500000000000000 781250000000000000 1953125000000000000 4882812500000000000 12207031250000000000 30517578125000000000 76293945312500000000 190734863281250000000 476837158203125000000 1192092895507812500000 2980232238769531250000 7450580596923828125000 18626451492309570312500 46566128730773925781250 116415321826934814453125

17179869184 42949672960 107374182400 268435456000 671088640000 1677721600000 4194304000000 10485760000000 26214400000000 65536000000000 163840000000000 409600000000000 1024000000000000 2560000000000000 6400000000000000 16000000000000000 40000000000000000 100000000000000000 250000000000000000 625000000000000000 1562500000000000000 3906250000000000000 9765625000000000000 24414062500000000000 61035156250000000000 152587890625000000000 381469726562500000000 953674316406250000000 2384185791015625000000 5960464477539062500000 14901161193847656250000 37252902984619140625000 93132257461547851562500 232830643653869628906250 582076609134674072265625

34359738368 85899345920 214748364800 536870912000 1342177280000 3355443200000 8388608000000 20971520000000 52428800000000 131072000000000 327680000000000 819200000000000 2048000000000000 5120000000000000 12800000000000000 32000000000000000 80000000000000000 200000000000000000 500000000000000000 1250000000000000000 3125000000000000000 7812500000000000000 19531250000000000000 48828125000000000000 122070312500000000000 305175781250000000000 762939453125000000000 1907348632812500000000 4768371582031250000000 11920928955078125000000 29802322387695312500000 74505805969238281250000 186264514923095703125000 465661287307739257812500 1164153218269348144531250 2910383045673370361328125

68719476736 171798691840 429496729600 1073741824000 2684354560000 6710886400000 16777216000000 41943040000000 104857600000000 262144000000000 655360000000000 1638400000000000 4096000000000000 10240000000000000 25600000000000000 64000000000000000 160000000000000000 400000000000000000 1000000000000000000 2500000000000000000 6250000000000000000 15625000000000000000 39062500000000000000 97656250000000000000 244140625000000000000 610351562500000000000 1525878906250000000000 3814697265625000000000 9536743164062500000000 23841857910156250000000 59604644775390625000000 149011611938476562500000 372529029846191406250000 931322574615478515625000 2328306436538696289062500 5820766091346740722656250 14551915228366851806640625

137438953472 343597383680 858993459200 2147483648000 5368709120000 13421772800000 33554432000000 83886080000000 209715200000000 524288000000000 1310720000000000 3276800000000000 8192000000000000 20480000000000000 51200000000000000 128000000000000000 320000000000000000 800000000000000000 2000000000000000000 5000000000000000000 12500000000000000000 31250000000000000000 78125000000000000000 195312500000000000000 488281250000000000000 1220703125000000000000 3051757812500000000000 7629394531250000000000 19073486328125000000000 47683715820312500000000 119209289550781250000000 298023223876953125000000 745058059692382812500000 1862645149230957031250000 4656612873077392578125000 11641532182693481445312500 29103830456733703613281250 72759576141834259033203125

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網站

http://www.bitman.name/media/pdf/upload/tables/regolari_1.txt

http://eskibars.com/projecteuler/finished/prob29data.txt

https://raw.githubusercontent.com/zma/fair-and-square/master/fair-and-sqaure.txt

http://www.cs.sjsu.edu/~mak/1401/powersOf2_printout.txt