A min problem of the year 2010 - Printable Version +- OMath! (http://math.elinkage.net) +-- Forum: Math Forums (/forumdisplay.php?fid=4) +--- Forum: Number Theory (/forumdisplay.php?fid=7) +--- Thread: A min problem of the year 2010 (/showthread.php?tid=27) A min problem of the year 2010 - elim - 03-08-2010 01:19 PM 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 $$10$$. RE: A min problem of the year 2010 - elim - 04-09-2010 03:13 PM 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 $$p \not | G$$ RE: A min problem of the year 2010 - elim - 05-01-2010 09:36 PM 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, 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