A min problem of the year 2010
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03-08-2010, 01:19 PM
Post: #1
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A min problem of the year 2010
Let \(0 < a,b,c \in \mathbf{N}^+\) such that \(a+b+c=2010\)
Find the smallest possible \(n \in \mathbf{N}^+\) so that \(a! \ast b! \ast c! = 10^n M\) where \(M\) is not divisible by \(10\). |
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04-09-2010, 03:13 PM
Post: #2
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RE: A min problem of the year 2010
I think that we need the following result:
If \(p\) is prime and \(m = \displaystyle{\sum_{k=1}^{\infty} \left[\frac{n}{p^k} \right]}\), then \(n! = p^m G\) for some \(G \in \mathbf{N}\) with \(p \not | G\) |
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05-01-2010, 09:36 PM
Post: #3
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RE: A min problem of the year 2010
There are 2712 solutions to make a!b!c! contain the smallest possible power of 10^m
where m = 493. Among them [624, 624, 762] makes a!b!c! the maximum. There are 336675 arrangements of \(a,b,c \in \mathbb{N}^+\) with \(a \le b \le c, \quad a+b+c=2010\) And so the probability to hit a solution here is about \(0.008\) [137, 624, 1249], [138, 623, 1249], [138, 624, 1248], [139, 622, 1249], [139, 623, 1248], [139, 624, 1247], [142, 619, 1249], [142, 624, 1244], [143, 618, 1249], [143, 619, 1248], [143, 623, 1244], [143, 624, 1243], [144, 617, 1249], [144, 618, 1248], [144, 619, 1247], [144, 622, 1244], [144, 623, 1243], [144, 624, 1242], [147, 614, 1249], [147, 619, 1244], [147, 624, 1239], [148, 613, 1249], [148, 614, 1248], [148, 618, 1244], [148, 619, 1243], [148, 623, 1239], [148, 624, 1238], [149, 612, 1249], [149, 613, 1248], [149, 614, 1247], [149, 617, 1244], [149, 618, 1243], [149, 619, 1242], [149, 622, 1239], [149, 623, 1238], [149, 624, 1237], [162, 599, 1249], [162, 624, 1224], [163, 598, 1249], [163, 599, 1248], [163, 623, 1224], [163, 624, 1223], [164, 597, 1249], [164, 598, 1248], [164, 599, 1247], [164, 622, 1224], [164, 623, 1223], [164, 624, 1222], [167, 594, 1249], [167, 599, 1244], [167, 619, 1224], [167, 624, 1219], [168, 593, 1249], [168, 594, 1248], [168, 598, 1244], [168, 599, 1243], [168, 618, 1224], [168, 619, 1223], [168, 623, 1219], [168, 624, 1218], [169, 592, 1249], [169, 593, 1248], [169, 594, 1247], [169, 597, 1244], [169, 598, 1243], [169, 599, 1242], [169, 617, 1224], [169, 618, 1223], [169, 619, 1222], [169, 622, 1219], [169, 623, 1218], [169, 624, 1217], [172, 589, 1249], [172, 594, 1244], [172, 599, 1239], [172, 614, 1224], [172, 619, 1219], [172, 624, 1214], [173, 588, 1249], [173, 589, 1248], [173, 593, 1244], [173, 594, 1243], [173, 598, 1239], [173, 599, 1238], [173, 613, 1224], [173, 614, 1223], [173, 618, 1219], [173, 619, 1218], [173, 623, 1214], [173, 624, 1213], [174, 587, 1249], [174, 588, 1248], [174, 589, 1247], [174, 592, 1244], [174, 593, 1243], [174, 594, 1242], [174, 597, 1239], [174, 598, 1238], [174, 599, 1237], [174, 612, 1224], [174, 613, 1223], [174, 614, 1222], [174, 617, 1219], [174, 618, 1218], [174, 619, 1217], [174, 622, 1214], [174, 623, 1213], [174, 624, 1212], [187, 574, 1249], [187, 599, 1224], [187, 624, 1199], [188, 573, 1249], [188, 574, 1248], [188, 598, 1224], [188, 599, 1223], [188, 623, 1199], [188, 624, 1198], [189, 572, 1249], [189, 573, 1248], [189, 574, 1247], [189, 597, 1224], [189, 598, 1223], [189, 599, 1222], [189, 622, 1199], [189, 623, 1198], [189, 624, 1197], [192, 569, 1249], [192, 574, 1244], [192, 594, 1224], [192, 599, 1219], [192, 619, 1199], [192, 624, 1194], [193, 568, 1249], [193, 569, 1248], [193, 573, 1244], [193, 574, 1243], [193, 593, 1224], [193, 594, 1223], [193, 598, 1219], [193, 599, 1218], [193, 618, 1199], [193, 619, 1198], [193, 623, 1194], [193, 624, 1193], [194, 567, 1249], [194, 568, 1248], [194, 569, 1247], [194, 572, 1244], [194, 573, 1243], [194, 574, 1242], [194, 592, 1224], [194, 593, 1223], [194, 594, 1222], [194, 597, 1219], [194, 598, 1218], [194, 599, 1217], [194, 617, 1199], [194, 618, 1198], [194, 619, 1197], [194, 622, 1194], [194, 623, 1193], [194, 624, 1192], [197, 564, 1249], [197, 569, 1244], [197, 574, 1239], [197, 589, 1224], [197, 594, 1219], [197, 599, 1214], [197, 614, 1199], [197, 619, 1194], [197, 624, 1189], [198, 563, 1249], [198, 564, 1248], [198, 568, 1244], [198, 569, 1243], [198, 573, 1239], [198, 574, 1238], [198, 588, 1224], [198, 589, 1223], [198, 593, 1219], [198, 594, 1218], [198, 598, 1214], [198, 599, 1213], [198, 613, 1199], [198, 614, 1198], [198, 618, 1194], [198, 619, 1193], [198, 623, 1189], [198, 624, 1188], [199, 562, 1249], [199, 563, 1248], [199, 564, 1247], [199, 567, 1244], [199, 568, 1243], [199, 569, 1242], [199, 572, 1239], [199, 573, 1238], [199, 574, 1237], [199, 587, 1224], [199, 588, 1223], [199, 589, 1222], [199, 592, 1219], [199, 593, 1218], [199, 594, 1217], [199, 597, 1214], [199, 598, 1213], [199, 599, 1212], [199, 612, 1199], [199, 613, 1198], [199, 614, 1197], [199, 617, 1194], [199, 618, 1193], [199, 619, 1192], [199, 622, 1189], [199, 623, 1188], [199, 624, 1187], [212, 549, 1249], [212, 574, 1224], [212, 599, 1199], [212, 624, 1174], [213, 548, 1249], [213, 549, 1248], [213, 573, 1224], [213, 574, 1223], [213, 598, 1199], [213, 599, 1198], [213, 623, 1174], [213, 624, 1173], [214, 547, 1249], [214, 548, 1248], [214, 549, 1247], [214, 572, 1224], [214, 573, 1223], [214, 574, 1222], [214, 597, 1199], [214, 598, 1198], [214, 599, 1197], [214, 622, 1174], [214, 623, 1173], [214, 624, 1172], [217, 544, 1249], [217, 549, 1244], [217, 569, 1224], [217, 574, 1219], [217, 594, 1199], [217, 599, 1194], [217, 619, 1174], [217, 624, 1169], [218, 543, 1249], [218, 544, 1248], [218, 548, 1244], [218, 549, 1243], [218, 568, 1224], [218, 569, 1223], [218, 573, 1219], [218, 574, 1218], [218, 593, 1199], [218, 594, 1198], [218, 598, 1194], [218, 599, 1193], [218, 618, 1174], [218, 619, 1173], [218, 623, 1169], [218, 624, 1168], [219, 542, 1249], [219, 543, 1248], [219, 544, 1247], [219, 547, 1244], [219, 548, 1243], [219, 549, 1242], [219, 567, 1224], [219, 568, 1223], [219, 569, 1222], [219, 572, 1219], [219, 573, 1218], [219, 574, 1217], [219, 592, 1199], [219, 593, 1198], [219, 594, 1197], [219, 597, 1194], [219, 598, 1193], [219, 599, 1192], [219, 617, 1174], [219, 618, 1173], [219, 619, 1172], [219, 622, 1169], [219, 623, 1168], [219, 624, 1167], [222, 539, 1249], [222, 544, 1244], [222, 549, 1239], [222, 564, 1224], [222, 569, 1219], [222, 574, 1214], [222, 589, 1199], [222, 594, 1194], [222, 599, 1189], [222, 614, 1174], [222, 619, 1169], [222, 624, 1164], [223, 538, 1249], [223, 539, 1248], [223, 543, 1244], [223, 544, 1243], [223, 548, 1239], [223, 549, 1238], [223, 563, 1224], [223, 564, 1223], [223, 568, 1219], [223, 569, 1218], [223, 573, 1214], [223, 574, 1213], [223, 588, 1199], [223, 589, 1198], [223, 593, 1194], [223, 594, 1193], [223, 598, 1189], [223, 599, 1188], [223, 613, 1174], [223, 614, 1173], [223, 618, 1169], [223, 619, 1168], [223, 623, 1164], [223, 624, 1163], [224, 537, 1249], [224, 538, 1248], [224, 539, 1247], [224, 542, 1244], [224, 543, 1243], [224, 544, 1242], [224, 547, 1239], [224, 548, 1238], [224, 549, 1237], [224, 562, 1224], [224, 563, 1223], [224, 564, 1222], [224, 567, 1219], [224, 568, 1218], [224, 569, 1217], [224, 572, 1214], [224, 573, 1213], [224, 574, 1212], [224, 587, 1199], [224, 588, 1198], [224, 589, 1197], [224, 592, 1194], [224, 593, 1193], [224, 594, 1192], [224, 597, 1189], [224, 598, 1188], [224, 599, 1187], [224, 612, 1174], [224, 613, 1173], [224, 614, 1172], [224, 617, 1169], [224, 618, 1168], [224, 619, 1167], [224, 622, 1164], [224, 623, 1163], [224, 624, 1162], [237, 524, 1249], [237, 549, 1224], [237, 574, 1199], [237, 599, 1174], [237, 624, 1149], [238, 523, 1249], [238, 524, 1248], [238, 548, 1224], [238, 549, 1223], [238, 573, 1199], [238, 574, 1198], [238, 598, 1174], [238, 599, 1173], [238, 623, 1149], [238, 624, 1148], [239, 522, 1249], [239, 523, 1248], [239, 524, 1247], [239, 547, 1224], [239, 548, 1223], [239, 549, 1222], [239, 572, 1199], [239, 573, 1198], [239, 574, 1197], [239, 597, 1174], [239, 598, 1173], [239, 599, 1172], [239, 622, 1149], [239, 623, 1148], [239, 624, 1147], [242, 519, 1249], [242, 524, 1244], [242, 544, 1224], [242, 549, 1219], [242, 569, 1199], [242, 574, 1194], [242, 594, 1174], [242, 599, 1169], [242, 619, 1149], [242, 624, 1144], [243, 518, 1249], [243, 519, 1248], [243, 523, 1244], [243, 524, 1243], [243, 543, 1224], [243, 544, 1223], [243, 548, 1219], [243, 549, 1218], [243, 568, 1199], [243, 569, 1198], [243, 573, 1194], [243, 574, 1193], [243, 593, 1174], [243, 594, 1173], [243, 598, 1169], [243, 599, 1168], [243, 618, 1149], [243, 619, 1148], [243, 623, 1144], [243, 624, 1143], [244, 517, 1249], [244, 518, 1248], [244, 519, 1247], [244, 522, 1244], [244, 523, 1243], [244, 524, 1242], [244, 542, 1224], [244, 543, 1223], [244, 544, 1222], [244, 547, 1219], [244, 548, 1218], [244, 549, 1217], [244, 567, 1199], [244, 568, 1198], [244, 569, 1197], [244, 572, 1194], [244, 573, 1193], [244, 574, 1192], [244, 592, 1174], [244, 593, 1173], [244, 594, 1172], [244, 597, 1169], [244, 598, 1168], [244, 599, 1167], [244, 617, 1149], [244, 618, 1148], [244, 619, 1147], [244, 622, 1144], [244, 623, 1143], [244, 624, 1142], [247, 514, 1249], [247, 519, 1244], [247, 524, 1239], [247, 539, 1224], [247, 544, 1219], [247, 549, 1214], [247, 564, 1199], [247, 569, 1194], [247, 574, 1189], [247, 589, 1174], [247, 594, 1169], [247, 599, 1164], [247, 614, 1149], [247, 619, 1144], [247, 624, 1139], [248, 513, 1249], [248, 514, 1248], [248, 518, 1244], [248, 519, 1243], [248, 523, 1239], [248, 524, 1238], [248, 538, 1224], [248, 539, 1223], [248, 543, 1219], [248, 544, 1218], [248, 548, 1214], [248, 549, 1213], [248, 563, 1199], [248, 564, 1198], [248, 568, 1194], [248, 569, 1193], [248, 573, 1189], [248, 574, 1188], [248, 588, 1174], [248, 589, 1173], [248, 593, 1169], [248, 594, 1168], [248, 598, 1164], [248, 599, 1163], [248, 613, 1149], [248, 614, 1148], [248, 618, 1144], [248, 619, 1143], [248, 623, 1139], [248, 624, 1138], [249, 512, 1249], [249, 513, 1248], [249, 514, 1247], [249, 517, 1244], [249, 518, 1243], [249, 519, 1242], [249, 522, 1239], [249, 523, 1238], [249, 524, 1237], [249, 537, 1224], [249, 538, 1223], [249, 539, 1222], [249, 542, 1219], [249, 543, 1218], [249, 544, 1217], [249, 547, 1214], [249, 548, 1213], [249, 549, 1212], [249, 562, 1199], [249, 563, 1198], [249, 564, 1197], [249, 567, 1194], [249, 568, 1193], [249, 569, 1192], [249, 572, 1189], [249, 573, 1188], [249, 574, 1187], [249, 587, 1174], [249, 588, 1173], [249, 589, 1172], [249, 592, 1169], [249, 593, 1168], [249, 594, 1167], [249, 597, 1164], [249, 598, 1163], [249, 599, 1162], [249, 612, 1149], [249, 613, 1148], [249, 614, 1147], [249, 617, 1144], [249, 618, 1143], [249, 619, 1142], [249, 622, 1139], [249, 623, 1138], [249, 624, 1137], [262, 499, 1249], [262, 624, 1124], [263, 498, 1249], [263, 499, 1248], [263, 623, 1124], [263, 624, 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